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CIHM/ICMH 

Microfiche 

Series. 


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Cc'lection  de 
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Technical  and  Bibliographic  Notes/Notes  techniques  et  bibliographiques 


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}S 


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illustrent  la  mAthode. 


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6 

1, 


THE  ELEMENTS 


OF  THE 


FOUR  INISTER  PLANETS 


AND  THE 


FUNDAMENTAL  CONSTANTS  OF  ASTRONOMY 


BY 


fV3 


S  - 


ij"^ 


SIMON   NEWCOMB 


Supplement  to  tbe  Americau  Ephemeris  and  Nautical 
Almanac  for  1897 


■-*  ♦  »•■ 


WASHINGTON 

GOVBRNMENT  PRINTING  OFFICE 
1895 


PREFACE. 


The  diversity  in  the  adopted  values  of  the  elements  and 
constants  of  astronomy  is  productive  of  inconvenience  to  all 
who  are  engaged  in  investigations  based  upon  these  quanti- 
ties, and  injurious  to  the  precision  and  symmetry  of  much  of 
our  astronomical  work.  If  any  cases  exist  in  which  uniform 
and  consistent  values  of  all  these  quantities  are  embodied  in 
iin  extended  series  of  astronomical  results,  whether  in  the 
form  of  ephemerides  or  results  of  observations,  they  are  the 
exception  rather  than  the  rule.  The  longer  this  diversity 
continues  the  greater  the  ditticulties  which  astronomers  of 
the  future  will  meet  in  utilizing  the  work  of  our  time. 

On  taking  charge  of  the  work  of  prejjaring  the  American 
Ephemeris  in  1877  the  writer  was  so  strongly  impressed  with 
the  inconvenience  arising  from  this  source  that  he  deemed  it 
advisable  to  devote  all  the  force  which  he  could  spare  to  the 
work  of  deriving  improved  v.alues  of  the  fundamental  elements 
and  embodying  them  in  new  tables  of  the  celestial  motions. 
It  was  expected  that  the  work  could  all  be  done  in  ten  years. 
But  a  number  of  circumstances,  not  necessary  to  describe  at 
present,  prevented  the  fulfillment  of  this  hope.  Only  now  is 
the  work  complete  so  far  as  regards  the  fundamental  constants 
and  the  elements  of  the  planets  from  Mercury  to  Jupiter  inclu- 
sive. The  construction  of  tables  of  the  four  inner  planets  is 
now  in  progress,  those  of  Jupiter  and  Saturn  having  already 
been  completed  by  Mr.  Hill.  All  these  tables  will  be  pub- 
lished as  soon  as  possible,  and  the  investigations  on  which 
they  are  based  are  intended,  so  far  as  it  is  practicable  to  con- 
dense them,  to  appear  in  subsequent  volumes  of  the  Astro- 
nomical Papers  of  the  American  Ephemeris.  As  it  will  take 
several  years  to  bring  out  these  volumes,  it  has  been  deemed 
advisable  to  publisL  in  advance  the  present  brief  summary  of 
the  work. 

Ill 


IV 


PREPAOE. 


The  author  feels  that  critical  examination  of  this  monograph 
may  show  in  many  points  a  want  of  consistency  and  conti- 
nuity. The  ground  covered  is  so  extensive,  the  material  so 
diverse  as  well  as  voluminous,  and  the  relations  to  be  investi- 
gated so  numerous,  that  no  conclusion  could  be  rt'ii(;hed  on 
one  point  which  was  not  liable  to  be  modified  by  subsequent 
decisions  upon  other  points.  The  author  trusts  that  the  diffi- 
culties growing  out  of  these  features  of  the  work,  as  well  as 
those  incident  to  the  administration  of  an  ofHce  not  especially 
organized  for  the  work,  will  afford  a  sufticient  apology  for  any 
defects  that  may  be  noticed. 

Nautical  Almanac  Office, 

U.  S.  Naval  Observatory,  January  7, 1895. 


CONTENTS. 


CHAPTEK     I.— GENKIIAL    OUTLINK    OK    THE    WOUK    OK    CUMPAHINU 
THE    OHSEKVATIOXS    WITH    THEOHY, 

J  1.  Reduction  to  the  standard  system  of  Right  Ascensions  and 
Declinations 

$    2.  Observations  nsed 

$  3.  Seniidiaineters  of  Mercury  and  Venus.— Table  for  defective 
illumination  of  Mercury  in  Right  Ascension 

$  4.  Tabular  places  from  Lkvekrieu's  tables.— Reduction  for 
masses  used  by  Lk VEimiKii 

^    5.  Coniparisoim  of  observations  and  tables 

$    6.  Equations  of  condition. — Method  of  formation 

$  7.  Method  of  determining  the  secular  variations  and  the  masses 
of  Venus  and  Mercury  independently 

$  8.  Method  of  introducing  the  results  of  observations  on  transits 
of  Venus  and  Mercury ;  separate  solutions,  A  from  meridian 
observations  without  transits;  Ji,  including  both  meridian 
observations  and  transits 


1 
1 


6 

8 
8 

10 


13 


Chaptek  II.— Discussion  and  results  ok  observations  of 

THE  Sun. 

$  9.  Method  of  treating  observed  Right  Ascensions  of  the  Sun.- 
Expression  of  errors  of  observed  Right  Ascension  as  error 
of  longitude 

$  10.  Treatment  of  observed  Declinations  of  the  Sun.— Formation 
of  equations  of  condition  for  the  corrections  to  the 
obliquity  and  to  the  Sun's  absolute  longitude 

$  11.  Formation  of  equations  froui  observed  Right  Ascensions  of 
Sun 

$  12.  Solution  of  equations  from  Right  Ascensions  of  the  Sun.— 
Tabular  exhibit  of  results  of  observations  of  the  Sun's 
Right  Ascensions  at  various  observatories  during  different 
periods 

$  13.  Mass  of  Venus,  derived  from  observations  of  the  Sun's  Right 
Ascension 

$  14.  Discussion  of  corrections  to  the  Right  Ascensions  of  the  Sun 
relative  to  that  of  the  stars 

V 


15 

16 
17 

20 
24 
25 


VI 


CONTENTS. 


Page. 

$  15.  DiHcussion  of  corroctious  to  the  eccentricity  and  perihelion 

of  the  Kiirth'a  orltit 27 

^  10.  KeHiiltH  of  ohservod  DeclinationH  of  tlio  Sun.— Exhihit  of 
individual  correctionH  to  t\w  abNoluto  longitude  and  the 
obli(|uity  of  tlio  ecliptic  at  the  ditlerent  obaervutoriea 
during  different  pcrioda 29 

^  17.  DiscuHBion  of  the  observed  corrections  to  the  Sun's  absolute 

longitude 32 

$  18.  Discussion  of  the  observed  corrections  to  the  obliquity  of 

the  ecliptic 33 

^  19.  Ktlcctof  refraction  on  the  obli(|uity;  special  investigation  of 
tlie  secular  change  of  obli(iuity  as  derived  iVom  observa- 
tiouH  of  the  Sun 35 

^  20.  Concluded  results  for  the  obliciuity,  and  its  socniar  varia- 
tion    39 

(  21.  Summary  of  results  for  the  corrections  to  the  elements  of  the 
Earth's  orbit  and  their  secular  variations  as  derived  from 
obsorvatio^js  of  the  Huu  alone 41 


CiiAPTKH    III, — Results    of   obsehvatio.ns    ok    thk    plankts 
MEiicrHY,  Vknus,  and  Maks. 


lit 


J  22.  Elements  adopted  for  correction 43 

$  23.  Introduction  of  the  corrections  to  the  masses  of  Venus  and 

M       iry 45 

$  24.  Ic  jtion  of  the  errors  of  absolute  Right  Ascension  and 

jiations  of  the  standard  stars 46 

$  25.  Introduction  of  the  corrections  to  the  secular  variations. — 
Method  of  forming  the  normal  equations  by  periods  so  as 

to  include  the  correction  to  the  secular  variation 49 

$  26.  Dates  and  weights  of  the  equations  for  the  various  periods.  52 

$  27.  Unknown  qnantities  of  the  equations. — Factors  for  changing 
corrections  of  the  unlinown  quantities  into  corrections  of 

the  elements 55 

$  28.  Table  of  the  values  of  the  principal  coefflcients  of  the  normal 

equations .56 

$  29.  Order  of  elimination 57 

$  30.  Treatment  of  meridian  observations  of  Mercury. — Effect  of 
want  of  approximation  in  the  coefficients  of  the  equations 

of  condition 58 

$  31.  Introduction  of  the  equations  derived  from  observed  tran- 
sits of  Mercury 61 

$  32.  Solntion  of  the  equations  for  Mercury 65 

$  33.  Systematic  discordances  among  the  observed  Right  Ascen- 
sions of  Mercury  in  different  points  of  its  relative  orbit. .  66 


CONTENTS. 

$  34.  Compariion  «tf  the  resnlts  derived  from  meridian  olmerva- 
tioiiH  of  Mercury  with  tliose  derived  from  trannitH  over  the 
Sun's  disk  

$  36.  Treatment  of  meridian  observations  of  Venus 

^  36.  lleHults  of  observed  transits  of  Venus 

^  37.  liquations  derived  from  observed  transits  of  Venus 

^  3H.  Solutions  of  the  ei|uations  from  Venus 

$  39.  Comparison  of  the  results  of  meridian  observations  of  Venns 
with  those  of  transits 

$  40.  Solution  of  the  equations  for  Mars. — Ine(|nality  of  long 
period  in  the  mean  longitude  and  perihelion,  indicated  by 
observations  

$  41.  Reduction  from  the  o<iuator  to  the  ecliptic 


VII 

I'RgO. 


69 
70 
70 
75 
70 


76 


77 
79 


Chapter  IV.— Comhination    of   tiik    i-KKCKumti    kesi-i.th   to 

OHTAIN  THE  MOST  PROUAHI.K  VAU'KS  Ol'  THE  EI.EMENTH 
AND  OK  TIIEIK  SKCULAH  VAUIATiONS  FKO.m'  OB8EKVA- 
TION8  ALONE. 

$  42.  Moditications  of  the  canons  of  least  8(iuares 81 

$  43.  Relative  precision  of  the  two  methods  of  determining  the 

elements  of  the  Earth's  orbit 86 

^  4 ' .  Concluded    secular    variations   of   the   solar   elements,    as 

derived  from  observations  alone 87 

^  45.  Common  error  of  the  standard  declinations 89 

$  46.  Definitive  secular  variations  of  all  the  elements  from  obser- 
vations alone. — Matrices  of  the  normal  equations  for  the 

secular  variations. — Tabular  statement  of  results 90 

^  47.  Definitive  corrections  to  the  solar  elements  for  1850 95 


Chapter  V. — Masses  ok   the  planets  derived   hy  methods 

INDKPKNDKNT    OK    THE    SECl'LAR    VARIATIONS,    WITH    THE 
RESULTING   COMPUTED   SECULAR    VARIATIONS. 

$  48.  Plan  of  discussion 97 

$  49.  Mass  of  Jupiter;  general  combination  of  results 9V 

$50.  Mass  of  Mars. — Prof.  Hall's  value  adopted 99 

$  51.  Mass  of  the  Earth,  derived  from  the  preliminary  value  of  the 

solar  parallax 99 

$  52.  Mass  of  Venus,  derived  from  periodic  perturbations 101 

$  53.  Ms.8s  of  Mercury,  from  various  sources 102 

$  54.  Theoretical  values  of  the  secular  variations  for  1850 106 


VIII 


r<»NTKNTS. 


Page. 


CiiAiTRR  VI.— Examination  <>k    iiypothkskh  anp  dp,tekmina> 

TION  ((K  TIIK  MAMSK.S  l«Y  Wllltll  TIIK  nKVIATKlNrt  OK  TIIK 
SRCn.Alt  VAItlAI'IONM  KIIOM  TIIKIK  TIIKOKKTICAI.  VALIIKS 
MAY   I»K   KXPI,AINKI>. 

« 

^  55.  ('oinpariHon  of  th«  obstrved  and  thfor(>ti<al  Hocular  varia- 
tions   

$  '>(».  IlypothoNiH  of  iionHpliericity  of  the  <>quiputoiitial  surfaces 
of  tJ:o  Snii 

^  Til.  HypotliBHis  of  an  intrain«Tciiriiil  ring 

$  .58.  ITypotliesiN  of  an  extended  niass  of  diffused  matter,  like  that 
which  rotlects  tho  /.odiacal  light 

$  59.  Hypothesis  of  a  ring  of  planets  outside  t1i«  orhit  of  Mer- 
cury.— Kh'Uients  of  such  a  ring. — This  hypothesis  the  only 
o\w  which  represents  the  observations,  but  too  im))robable 
to  he  accepted 

^  60.  Examination  of  the  question  whether  tiie  excess  of  motion 
of  the  perilielion  of  Mars  may  be  due  to  the  action  of  the 
zone  of  minor  planets 

$  61.  Hypothesis  that  gravitation  towari  clie  Sun  is  not  exactly 
as  the  inverse  sq-uaro  of  the  distance 

$62.  Degree  of  precision  with  which  the  theory"  of  the  inverse 
S(piare  is  established 

$  63.  Determination  of  the  masses  which  will  best  represent  the 
observed  secular  variations  of  the  eccentricities,  uodes, 
and  inclinations 

$  6-t.  Preliminnry  adjustment  of  the  two  sets  ot  masses. — Result- 
iug  valuta  of  the  solar  parallax 

ClIAPTKK    VII. — VaI.T'ES     of    THE     PRINCIPAL    CONSTANTS    WHICH 
DK1'KNI>    UP«)N    THE   MOTION    OF   TIIK  EaIMII. 

^  65.  The  precessional  constant 

^  66.  The  constant  of  nutation,  derived  from  observations 

%  67.  Relations  between  the  constants  of  precession  and  nutation 
and  the  quantities  on  which  they  depend 

$  68.  The  mass  of  the  Moon  from  the  observed  constant  of  nuta- 
tion   

$  69.  The  constant  of  aberration 

$  70.  The  values  of  this  constant,  derived  from  observations 

^  71.  The  lunar  inequality  in  the  P^arth's  motion 

§  72.  The  solar  parallax  derived  from  the  lunar  inequality 

$  73.  Values  of  the  solar  parallax  derived  from  measurements  of 
Venus  on  the  face  of  the  Sun  during  the  transits  of  l674 
and  1882,  with  tbe  heliometer  and  photoheliograph 

$  74.  The  solar  parallax  from  observed  contacts  during  transits  of 
Venus 


lOJ) 

111 
112 

115 


116 


116 


118 


119 


121 


122 


124 
129 

131 

132 
133 
135 
139 
142 


143 
145 


CONIT-NTS. 

'i  7D  Soliir  |»iiralliix  from  tlir  (>li.s<'ivnl  niHMtant  ofulHTriition  jiml 
niiMiHiiriMl  vt'lix'ity  <»!'  liylit 

^    7().  Solur  parallax  from  tho  parallactic  in<M]iiality  of  tlii<  Moon 

S^  77.  Solar  parallax  from  olmervatioiiti  of  tho  minor  plaiu-ts  with 
the  h(*liuiiii-tvr 

^  78.  Komarks  on  (U'tvrminationH  of  tho  parallax  which  iiro  not 
iiHeil  in  the  proHent  (liwcnsHion.  — Krrors  ariNioK  from  dlf- 
fert-nccs  of  color 

Chai'TEH  VIII.— DisrissioN  oi'  uksui.ts  kok  tiik  soi.aii  ••ahai.- 

I.AX    AND   TIIK    MASSK8   <»K   TIIK    TIIUKK    INNKI!    I'I.aNKTS. 

$    7!t.  S«"paratc  vahu^H  of  tho  solur  parallax,  and   thoir  gonorul 

moan 

$    80.  KediHcnNHion  of  tho  motion  of  tho  nodo  of  Vonnn 

sS    81.  I'oHHildoHysteinatic  orrorH  in  dt^termi  nations  of  tho  parallax. 

V^    82.   UoviMod  list  of  dotorminatioiiR 

,1    83.  Dcfinitivo  adjustmont  of  tho   nnihsuH  of  tho    three   inner 

planets 

^    81.  TosHihle  canHON  of  tho  ohsorvod  discoi dances 

S^    8,">.  Adopte<l  valncH  of  thn  donhtfnl  )|na)ititi:'s 

'^    8H.  Kearing  of  future  determinations  on  ^ho  <|iu-stion. 

Chai'tkk  IX.— DKHivA.n«»N  or  i<k,siji,ts, 

^    87.  Ulterior  corrections  to  the  motions  of  tho  perihelion  and 

mean  longitude  of  Morcnry 

^    HS.  Dofmitivo  elements  of  tho  fonr  inner  planets  for  the  e|>och 

18.">0,  as  inferred  from  all  the  dojta  of  ohsorvation . 

^    89.  Definitive  values  of  the  secular  vari;i.tioiis 

^  .!K).  Secular  accoleration  of  the  mean  motions 

^    lU.  The  measure  of  time 

^    J)2.  The  "onstant  of  aberration 

^    }t3.  The  mass  of  tho  Moon 

^    94.  The  parallactic  inequality  of  the  Moon 

\S    i)'>.  The  centimoter-second  system  of  astrorkondcal  uu  its 

^S    9(5.  Masses  of  tho  Karth  and  Moon  in  centimoter-second  units.. 

SS    97.  Parallax  of  the  Moon 

^    98.  Mass  and  parallax  of  the  Sun 

(J    99.  Constant  of  nutation,  and   mechanical  ellipticity  of  the 

Earth 

$  100.  Precession  

^  101.  Obliqu.ty  of  the  ecliptic 

^  102.  Relative  jtositions  of  the  eqnator  and  the  eclii»tic  at  ditter- 

eut  epochs  for  reduction  of  places  of  stars  and  planets  .. 


IX 

I'agtt. 

117 
118 

154 


15« 
l.'>9 
Uil 

im 

1«>8 
173 
173 
17r. 


178 

179 

182 
180 
188 
188 
189 
190 
190 
191 
193 
194 

195 
196 
196 

197 


ELEMENTS  AND  CONSTANTS. 


CHAPTER  1. 

GENERAL  OUTLINE    OF  THE  WORK    OF  COMPARING  THE 
OBSERVATIONS  WITH  THEORY. 

1.  In  logical  order,  the  first  step  in  tlie  work  consists  in  the 
reduction  of  observed  positions  of  the  Sun  and  planets  to  a 
uniform  equinox  and  system  of  declinations. 

The  adopted  standard  of  Right  Ascensions  was  that  origi- 
nally worked  out  in  my  paper  on  the  Right  Ascensions  of  the 
fundamental  stars,  found  in  an  appendix  to  the  Washington 
Observations  for  1870,  and  extended  to  a  fundamental  system 
of  time  stars  in  the  catalogue  published  in  Vol.  1  of  the  Astro- 
nomical Papers  of  the  American  Ephemeris.  This  system 
CGi.icides  closely  with  that  of  the  Astronomische  Gesellschaft 
and  the  Berliner  Jahrbuch,  about  the  epoch  1870,  but  the  cen- 
tennial proper  motion  is  greater  by  about  C.OS. 

In  Declinations,  the  adopted  standard  was  that  of  Boss, 
which  has  been  used  in  the  American  Ephemeris  since  1881, 
and  on  which  is  based  the  catalogue  of  zodiacal  stars  just 
referred  to.  But  as  Declinations  generally  are  not  immediately 
referred  to  fundamental  stars,  the  method  of  reducing  obser- 
vations to  this  system  in  Declination  was  not  entirely  uniform. 

Ohserrations  used. 

2.  The  following  is  a  general  statement  of  the  observati(ms 
used,  and  the  extent  to  which  t_  ay  were  corrected,  or  re-re- 
duced. 

Oreentcich. — Dr.  Auwers  courteously  supplied  me  with  the 
•Qsults  of  his  re-reduction  of  Bradley's  observations  both  of 
the  Sun  and  planets.  From  the  beginning  of  Maskylene's 
work  until  1835,  the  Greenwich  observations  were  completely 
re-reduced,  utilizing,  so  far  as  possible,  Aiby's  reductions.  The 
5690  N  ALM 1  1 


i*   ! 


GENERAL  OUTLINE. 


12 


data  necessary  for  these  observations  were  discussed  in  Prof. 
Saffokd's  paper,  Vol.  ii,  pt.  ii,  wliicli  paper  was  prepared 
for  this  purpose.  In  the  case  of  the  Greenwich  observations 
from  1835  onward,  it  was  deemed  sufficient  to  apply  constant 
corrections  to  the  Kight  Ascensions,  determined  from  time  to 
time  by  comparisons  of  the  adopted  Eight  Ascensions  with 
the  standard  ones.  In  the  case  of  the  Declinations,  Boss's 
special  tables  were  used,  but  in  the  later  years  it  was  judged 
sufficient  to  apply  the  constant  correction  necessary  for  reduc- 
tion to  Boss's  standard. 

Palermo. — PiAzzi's  observations  of  the  Sun  and  Planets  ware 
completely  re-reduced,  the  zero  point  of  his  instrument  being- 
determined  from  the  observed  Declinations. 

Paris. — LeVerkieb's  reduction  of  the  Paris  observations 
from  1801  onward  was  made  use  of,  applying  the  corroction 
necessary  to  reduce  the  results  to  the  adopted  standard. 

Kon'ujHberg. — Bessel's  clock  corrections  were  individually 
corrected  by  the  new  positions  of  the  fundamental  stars,  so 
that  practically  the  Eight  Ascensions  may  be  considered  as 
completely  re-reduced. 

In  the  case  of  the  other  observatories,  it  was  deemed  suffi- 
cient to  determine,  by  a  comparison  of  the  adopted  or  of  the 
concluded  Right  Ascensions  and  Declinations  of  the  funda- 
mental stars  with  the  stan<lard  catalogue,  what  common  cor- 
rections were  necessary  for  reduction  to  the  standard.  When, 
however,  the  period  was  covered  by  Boss's  tables,  the  correc- 
tion which  he  gives  as  varying  with  the  Declination  was  ap- 
plied. After  more  mature  consideration,  I  am  inclined  to  think 
it  would  have  been  better  to  apply  a  constant  correction  to  the 
Declinations  in  every  case,  except  those  where  the  change 
with  the  Declination  was  quite  large. 

Although  these  processes  were  somewhat  heterogeneous,  it 
is  believed  that  the  main  object  of  referring  the  Declinations 
to  a  system  of  which  the  error  would  be  a  uniformly  varying 
quantity  was  fairly  well  attained.  The  subsequent  deternii- 
nation  of  this  error  both  in  Eig'jt  Ascension  and  Declination 
is  a  necessary  part  of  the  work. 


12 


3]  OBSERVATIONS  USED.  8 

The  following  is  a  list  of  the  observatories  whose  observa- 
tions of  the  Sun  and  Planets  wwe  included  in  the  Avork : 

Greenwich -—  1 750-1892 

Palermo 1791-1813 

Paris - 1801-1889 

Konigsberg 1814-1845 

Dorpat -- --  1S23-1838 

Cambridge 1828-1844 

Berlin 1838-1842 

Oxford,  Radcliffe 1840-1887 

Pulkowa --- 1842-1875 

Washington 1846-1891 

Leiden 1863-1871 

Strassburg 1 884-1887 

Cape  of  Good  Hope 1884-1890 

The  number  of  the  meridian  observations  of  the  Sun,  and 
of  the  planets  Mercury,  Venus,  and  Mars,  actually  included  in 
the  work  is  approximately  as  follows: 

The  Sun _._ 40, 176 

Mercury 5.421 

VenuR 12,  'iq 

VuTS 4  114 

Total 62,030 

Semidiameterx  of  Mercury  and  Venus. 

.'{.  The  reduction  of  the  semidiameter  of  the  planets  was  a 
point  to  which  special  attention  was  given.  In  the  case  of 
Mercury,  the  adopted  semidiameter  at  distance  unity  was  3".34. 
The  vahies  adopted  by  the  various  observatories  in  reducing 
their  observations  varied  so  little  from  this  that  in  cases  where 
the  original  reductions  were  accepted  no  correction  was  applied 
for  the  dift'erence.  So,  also,  when  the  observers  applie<l  a  cor- 
rection for  reducing  the  observed  center  of  light  to  the  actual 
center  of  the  planet,  no  rsvision  of  this  reduction  was  made. 
Such  was  supposed  to  be  the  case  with  the  Paris  observations. 
When  the  published  Kight  Ascension  was  that  of  the  center 
of  light  simply,  a  reduction  to  the  true  center  was  computed 
by  the  empirical  formula  used  in  the  Washington  observations. 
If  we  put  i  for  the  angle  between  the  Earth  and  Sun  as  seen 
from  the  planet,  then  1  +  cos  i  will  represent  the  friictiou  of 


'    I 


GENERAL   OUTLINE. 


[3 


I 'I 

1    ' 


the  apparent  transverse  diameter  of  the  plauet  that  is  illu- 
minated by  the  Sun.  It  was  assumed  that  when  the  illumina- 
tion was  such  that  the  thickness  of  the  crescent  approached 
zero,  the  point  observed  would  be  two-thirds  of  the  way  from 
the  center  of  the  planet  to  the  limb,  and  that  when  the  planet 
was  dichotomized  the  center  of  observation  would  be  five- 
twelfths  of  the  Avay  from  the  center  to  the  limb.  These  con- 
ditions, with  the  added  one  that  when  the  phiuet  was  fully 
illuminated  the  correction  should  vanish,  suggested  the  em- 
ployment of  the  formula 

Correction  =  semidiameter  x  (1-cos  /)J5-f  cos  i) 

U 

This  correction  was  to  be  multiplied  by  the  sine  or  cosine  of 
the  angle  which  the  line  of  cusps  made  with  the  meridian  to 
reduce  it  to  Right  Ascension  and  Declination  respectively. 

The  correction  being  practically  the  same  whenever  the 
Earth  and  planet  return  to  the  same  positions  in  anomaly,  it 
is  possible  to  embody  it  in  a  table  of  two  arguments,  one 
depending  on  the  longitude  of  the  Earth,  the  other  on  that  of 
the  planet.  Actually,  however,  the  table  was  arranged  in  a 
more  convenient  form,  in  which  one  argument  is  the  date  at 
which  Mercury  last  passed  perihelion,  and  the  other,  its  mean 
anomaly.  Owing  to  the  importance  which  this  correction  may 
assume,  a  partial  transcript  of  the  table  actually  employed  for 
the  reduction  in  Right  Ascension  is  given  on  the  next  page. 
Read  horizontally,  the  numbers  show  the  corrections  of  the 
argument  through  one  revolution  of  the  planet.  Vertically, 
the^  may  be  regarded  as  giving  the  successive  corrections  corre- 
sponding to  any  one  position  of  the  planet,  while  the  Earth 
goes  through  a  complete  revolution.  The  table  as  actually 
used  extended  to  every  10°,  but  the  values  for  every  G0°  of 
mean  anomaly  will  suffice  to  show  the  general  magnitude  of 
the  correction. 

The  correction  to  the  Declination  was  embodied  in  a  similar 
table,  which  it  is  not  deemed  necessary  to  print  at  present. 

In  the  case  of  Venus,  it  seems  scarcely  possible  to  decide 
upon  a  value  of  the  semidiameter,  or  a  law  of  its  apparent 
change,  which  should  apply  to  all  parts  of  the  orbit.    After  a 


3] 


SEMIDIAMETERS  OF  MEKCURY  AND   VENIJS. 


careful  examination  of  the  data,  it  was  decided  to  reduce  all 
the  observations  with  the  .semidiameter 

when  made  with  modern  instrum  ^nts,  and  to  use  a  value  0".3 
greater  in  earlier  observations.    The  actual  reductions  of  all 

Correction  for  defective  illumination  of  Mercury  in  R.  A. 
Arguments:  Date  of  perihelion  pasHage  at  side,  and  mean 
anomaly  "^"  at  top. 


.c^ 


Jan. 


Feb. 


Mar. 


Apr. 


May 


June 


July 


Sept. 


Oct. 


Nov. 


Dec 


Jan. 


o . 

ID. 
20. 

30. 
9- 

«9- 
I  . 

II  . 
21 . 

31 

10. 

20. 

30 

10. 

20. 

30. 

9. 

«9. 

9.g. 

9 


«9 
29. 

Aug.  8 . 
18. 
28. 


7 

17 
27 

7 

17 
27 

6 

16. 
26, 

6 

16. 

26 
5 


60° 


120° 


180° 


+•'9 
.16 

.14 
.  12 

.  10 

.08 
.06 
.05 
.04 
.03 
.02 
.02 
+.01 
.00  I 
.00  ! 

.00 

.00  1 

.00  j 

-.01  i 

-.01  I 

— .01  I 
— .  02  i 
—.03 
—.04  , 
-.05 

—.06 
-.07  i 
— .09  ! 
— .  II   i 

— .12   j 

—.14  I 
—.16  ! 
— .  18, 


+  .20 

4-.  18 


s 

—.16 
—.18 

— .  21 
—  .19 
-17 


+  .16 
.16 

•»5 
.14 

.  12 

.  10 
.08 
.06 
•05 
.04 

•03 
.02 
.01 
.01 

+.01 
.00 
.00 
.00 
.  00 

.00 
— .01 
—.01 
— .02 
— .02 

-03 
—.04 
—.06 
—.08 
— .  10 

— .  12 

-.15 
— ■«7 


s 

.07 
.09 
.  II 
•»3 
•«5 
.18 
.  21 
.24 
.26 


+•23 
.  20 
.18 

•'5 
.  12 

.  10 
.09 
.07 

•OS 
.04 

•03 
.02 

.01 

.01 

+.01 

,00 
.00 
.00 
.00 
.00 

—.01 
—.01 
— .01 
— .02 
-03 

-.OS 
—.06 
—.08 


—  03 
-.04 
-OS 
—.06 
—.08 

— .  10 
— .  12 

-•'5 

-.  18 
— .  20 

— .22 


+.24 
.  22 
.  20 

.17 
•14 
.  12 
.09 
.07 

•OS 
.04 

•03 
•03 
.02 

.02 
+.01 
.00 
.00 
.00 

.00 
.00 
.00 
.00 
--.01 

— .01 
— .02 
—  03 


240° 

! 
300°  i 

i 

s 

1 

1  —.01 

.00 

1  -.01 

.00 

—.02 

.00 

-■03 
—.04 

.00 
—.01 

-OS 

—.06 
—.08 

— .  10 

— .  12 

—.01 
—.02 

—  03 
—.04 
-.06 

-•15 
-.18 
—  .21 

-.07 
—.09 
— .  II 

—•17 
— .  12 

—•13 
—.16 

.16 

—.18 
—.20 
— .  20 

'5 

— .  20 

.13 

.  II 
.09 
.07 
.06 
•OS 

+.  16 
.16 
.16 
.14 
•13 

.04 
.02 
.02 
+.01 
.00 

'.  II 

.09 
.07 

•OS 
.04 

.00 

!   .00 

•03 
.02 

.00 

H-.oi 

.00 

.00 

.00 

.00 

— .01 

.00! 

—.01 

.00 

—.01 

.00  1 

i 

360° 


s 

+•03 

.  02 
.02 

4".  01 
.  00 

.00 
.00 
,co 
.00 
— .01 

— .01 
— .01 
— .  02 

—•03 
—.04 


OS 
,06 
,07 
,09 
.  II 

.  12 

14 
,16 

18 


+.  20 
.18 
.16 

•13 
.  II 
.09 
.07 
.06 

•OS 
.04 

+.03 


6 


GENERAL   OUTLINE. 


the  principal  series  of  observations  were  corrected  to  this  value 
of  tlie  element  in  question. 

Observiitions  of  the  estimated  center  of  Venus,  when  made 
more  than  one  hundred  days  from  superior  conjunction,  were 
rejected  altogether;  wlien  made  within  that  limit,  the  point 
observed  was  assumed  to  be  the  center  of  gravity  of  the  illu- 
minated portion  of  the  disk,  considered  as  a  plane  ligure,  and 
the  necessary  reduction  to  the  center  was  always  applied. 

A  similar  correction  was  applied  to  observations  of  the  esti- 
mated center  of  Mars.  The  Paris  results,  after  18.30,  and  the 
later  (xreenwich  and  Washington  results,  are  published  with 
the  reduction  for  center  of  light  already  applied,  and  in  these 
cases  the  published  corrections  were  not  changed. 

Tabular  places. 

4.  The  tabular  elements  of  the  pl;inets  adopted  for  correc- 
tion were  those  of  Leverrier's  tables.  These  tables  having 
been  continuously  used  in  Astronomical  Ephemerides  since 
18G4,  it  was  judged  more  convenient  to  adopt  the  theory  on 
which  they  were  based  as  the  iirovisional  one  to  be  corrected 
than  it  was  to  construct  a  new  provisional  theory.  As  the  tables 
in  their  original  form  are  extremely  cumbrous  to  use,  the 
theory  was  partially  reconstructed  by  making  manuscript 
tables  of  the  principal  perturbations,  which  were,  however, 
carried  only  to  tenths  of  seconds.  With  these  tsibles  the 
places  of  the  planets  were  computed  for  dates  previous  to  18G4. 

As  places  of  the  Sun  were  necessary  not  only  for  direct  com- 
parison with  observations  of  the  Sun,  but  also  for  the  geocen- 
tric places  of  the  jdanets,  an  ephemeris  of  the  Suu  s  longitude 
and  radius  vector  was  prepared  for  the  entire  period  1750-1864 
to  every  fifth  day,  the  lunar  perturbation  being  omitted  and 
afterward  applied  for  each  date  when  required. 

The  method  of  deriving  the  final  tabular  places  varied  with 
circumstances.  When  there  was  no  accurate  ephemeris  avail- 
able for  comparison,  which  was  the  case  before  1830,  it  was 
necessary  to  compute  a  completely  independent  set  of  tabular 
geocentric  places.  Sometimes  these  places  were  computed  for 
the  moment  of  the  individual  observations,  but  more  generally, 
when  the  observations  occurred  in  groups,  an  ephemeris  was 


4] 


TABULAR   PLACES. 


computed  in  order  that  the  work  might  be  cliecked  by  diflF^r- 
ences.  After  1830  it  was  common  to  compute  an  ei»hemeris 
for  intervals  of  three,  five,  or  ten  days,  thus  deriving  the  cor- 
rections necessary  to  reduce  the  published  ephemerides  of  the 
Berliner  Jahrbuch  or  of  the  Nautical  Almanac  to  those  derived 
from  Leverriek's  tables. 

Until  this  plan  was  mapped  out,  and  work  well  in  progress 
upon  it,  it  was  not  noticed  that  the  planetary  masses  adopted  in 
Leverrieu's  tables  were  so  diverse  that  corrections  to  reduce 
the  geocentric  places  to  a  uniform  system  of  masses  would  be 
necessary.  Although  theoretically  the  necessary  reductions 
were  very  simple,  I  can  not  but  feel  that  the  application  of 
such  corrections  involves  more  or  less  doubt  aiul  uncertainty, 
and  that  it  would  have  been  better  to  have  constructed  pro- 
visional tables  based  on  uniform  masses  quite  independent  of 
those  of  Leverrier. 

In  Annah's  tie  V Ohscrvatoire  de  Paris,  Vol.  ii,  Leverrier 
gives  the  following  values  of  the  masses  used  by  him  as  the 
basis  of  his  provisional  theory: 


1 
Mercury 3000 000 - '^^^ ^^^^ ^^^  '  ' 

1 

^enus ioi^sSr  =•<><><>  002  4885 

Earth 354936   =-000  002  8174 

Mars 2680337  =  '^^^^  ^^  ^^^  ^^'^ 

The  following  .ible  shows  the  factors  by  which  these  masses 
were  multiplied  in  the  cases  of  the  several  planets  in  Lever- 
RiER's  final  tables.  They  were  controlled  by  induction  from 
the  numbers  of  the  tables  themselves,  the  result  of  which  was 
found  in  all  cases  to  agree  with  the  statements  i;i  the  introduc- 
tion to  the  tables. 

In  the  last  line  of  the  table  is  shown  the  factor  used  in  the 
present  provisional  theory. 


8 


GENERAL  OUTLINE. 


[5 


Mercury. 

Venus. 

Earth. 

Mars. 

In  tables  of — 

The  Sun 

I 

1.004 

I 

0.895 

Mercury  ..       .       

I 

I 

1.0026 

I 

Venus .. . 

I 

I 

Mars 

0-97S 

I 

Present  work 

I 

0.  8657 

As  in  the  actual  work  tlie  masses  of  Mercury  and  Venus 
were  to  be  determined  from  the  observed  periodic  perturba- 
tions which  they  produced,  it  was  necessary  that  the  perturba- 
tions produced  by  them  should  all  be  carefully  reduced  to  the 
adopted  standard.  The  reduction  was  less  necessary  in  the 
case  of  Mars,  but  was  carried  through  all  the  work  relating  to 
the  Sun. 

Comparison  of  observations  and  tables. 

5.  The  result  of  each  separate  observation  of  each  body  was 
compared  with  the  tabular  result  thus  derived.  The  residuals 
were  then  taken  and  divided  into  groups.  The  interval 
between  th«  extreme  dates  of  each  group  was  always  taken 
so  short  that  it  could  be  j^resumed  that  the  mean  of  all  the 
residuals  would  be  the  correction  for  the  mean  of  all  the  dates. 
The  j^eneral  rule  was  that  the  interval  should  not  e\ceed  four 
or  five  days  in  the  case  of  Mercury,  or  six  or  eight  days  in 
that  ot* Venus,  and  that  not  more  than  six  or  eight  observa- 
tions should  be  included  in  a  single  group.  In  taking  these 
means,  weights  were  assigned  to  the  results  of  each  observa- 
tory founded  on  the  discordance  of  its  residuals.  Then  to  each 
mean  a  weight  was  again  assigned  equal  to  the  sum  of  the 
weights  of  the  individual  residuals  when  these  were  few  in 
number,  but  not  allowed  to  exceed  a  certain  limit,  how  great 
soever  might  be  the  sum  of  the  individual  weights. 

Equations  of  condition. 

6.  Each  meaji  result  thus  derived  formed  the  absolute  term 
of  an  equation  of  condition  for  correcting  the  tabular  elements. 
The  number  of  these  equations  was  as  follows: 

Equations. 

The  Sun 11,676 

Mercury  .._ 3,929 

Venus ._ _.     4,849 

Mars 1,597 


6] 


EQUATIONS  OF   CONDITION. 


9 


III  forming  the  equations  of  condition  from  observations  of 
the  planets,  I  adopted  the  system  suggested  in  the  introduc- 
tion to  Vol.  I  of  these  publications,  namely,  the  determination 
of  the  solar  elements  not  only  from  observations  of  the  Sun 
itself,  but  from  observations  of  each  of  the  planets.  The  reason 
for  this  course  is  quite  simple  and  obvious.  An  observation  of 
the  position  of  a  planet  as  seen  from  the  Earth  is  the  exact 
equivalent  of  an  observation  of  the  Earth  as  seen  from  a 
planet,  and  thus  depends  equally  upon  the  elements  of  both 
orbits.  Hence,  whatever  elements  of  the  Earth's  orbit  could 
be  determined  by  observations  made  from  a  planet  can  equally 
be  determined  by  observations  made  upon  the  planet.  A 
strong  reason  for  proceeding  upon  this  plan  was  found  in  the 
very  large  errors,  both  accidental  and  systematic,  to  which 
observations  of  the  Sun  are  liable. 

The  advantages,  however,  have  not  proved  relatively  so 
great  as  were  anticipated.  The  eccentricity  and  perihelion  of 
the.  Earth's  orbit  come  out  in  the  solution  of  the  normal  equa- 
tions as  functions  of  those  of  the  planetary  orbit  to  so  great  an 
extent  that  their  weight  is  much  less  than  that  which  would 
correspond  to  independent  determinations  from  the  same  num- 
ber of  observations.  On  the  other  hand,  the  determination 
of  these  elements  from  observations  of  the  Sun  proved  to  be 
much  more  consistent  than  was  expected,  thus  indicating  a 
high  degree  of  precision. 

The  case  is  different  with  the  Sun's  mean  longitude  referred 
to  the  Stars.  Here  systematic  and  personal  errors  enter  so 
largely  that  the  results  from  Mercury  and  Venus  appear  to  be 
rather  more  reliable  than  those  from  the  Sun  itself.  In  the 
case  of  these  planets  it  fortunately  happens  that  the  weight  of 
the  result  derived  for  the  Sun's  mean  longitude  is  not  mate- 
rially diminished  by  the  uncertainty  of  the  corresponding 
element  of  the  planet,  the  errors  of  the  two  mean  longitudes 
being  nearly  separated  in  a  series  of  observations  equally  dis- 
tributed around  the  orbit. 

The  systematic  errors  in  observations  of  the  Sun  rendered 
it  unadvisable  to  determine  the  elements  of  the  Earth's  orbit 
from  observations  of  the  Sun  by  a  single  system  of  equations. 
The  solar  observations,  therefore,  were  classified  according  to 


10 


GENERAL   OUTLINE. 


[• 


the  observatory  wliero  made,  and  divided  into  periods  rarely 
exceeding  eiglit  years  in  length.  The  elements  are  separately 
derived  from  the  observations  of  each  i)eriod.  This  system  has 
the  advantage  of  eliminating  to  a  large  extent  the  injurious 
effect  of  systematic  and  personal  error  upon  the  eccentricity 
and  perihelion  of  the  Earth's  orbit,  and  also  enabling  us  to 
judge  of  the  precision  of  the  corrections  to  those  elements  by 
the  discor<lance  among  separate  results. 

Meridian  observations  of  the  Sun  and  Planets  are  referred 
to  the  fundamental  stars,  while  the  Kiglit  Ascensions  of  the 
latter  are  referred  to  the  equinox,  the  jiosition  of  which  has 
heretofore  depended  on  observacions  of  the  Sun.  The  adopted 
position  of  the  fundamental  stars  therefore  comes  in,  to  a  cer- 
tain extent,  as  the  basis  of  the  work,  and  the  constant  parts 
of  thf^ir  systematic  corrections  are  among  the  results  to  be 
derived. 

Thus,  in  the  case  of  the  equations  pertaining  to  the  three 
l)lanets,  the  following  corrections  were  introduced  as  unknown 
((uantitiea: 

Correction  of  the  mass  of  ^lercury  or  of  Venus. 

Corrections  to  the  elements  of  the  orbit  of  the  planet 
observed. 

Correction  of  the  obliquity  of  the  ecliptic. 

Corrct'tions  to  the  Sun's  mean  longitude,  eccentricity,  and 
longitude  of  perihelion. 

Common  corrections  to  the  a<k)pted  Kight  Ascensions  and 
Declinations  of  the  fundamental  stars. 

In  the  case  of  Mercury  an  adopted  hyi)othetical  correction 
of  the  ratio  of  the  radius  vector  of  the  planet  to  that  of  the 
Earth  was  also  included  in  the  equations,  although  little  doubt 
could  be  felt  that  the  true  v.'ilue  of  such  a  quantity  must  be 
zero.  The  reason  for  introducing  it  will  be  explained  here- 
after. 


Determinations  of  the  masses  and  secular  variations. 

7.  The  secular  variation  of  all  the  preceding  elements,  the 
mean  distances  excepted,  was  also  introduced  into  the  equa- 
tions from  observations  of  the  planets.  In  addition  to  the 
above  elements,  the  mass  of  Venus  appeared  in  the  equations 


7J 


MASHES   AND   SECULAU   VAHIATIONH. 


11 


derived  from  observatioiiH  of  tl>e  Sun,  Mercury,  and  Mur»,  and 
the  mass  of  Mercury  in  tlie  e(iuation8  derived  from  obser- 
vations of  Venus.  Tlie  coellicients  of  tlic  masses,  liowever, 
depended  wholly  upon  tlie  periodi*-  perturbations. 

Were  it  quite  certain  that  the  secular  variations  arise 
wholly  from  the  nuisses  of  the  known  planets,  the  masses 
could  of  course  be  derived  from  these  variations,  and  the  lat- 
ter would  appear  in  the  equations  of  condition  only  throuH;h 
the  mass  itself.  On  this  hypothesis  the  secular  variations 
would  not  appear  in  the  etpiations,  but  only  the  masses,  liut 
it  is  well  known  that  the  periheli(m  of  Mercury  is  subject  to  a 
secular  variation  which  can  not  be  accounted  for  by  any  ad- 
missible masses  of  the  known  disturbin;^'  jdanets.  The  same 
thing  may  well  be  true  of  the  secular  variations  of  the  other 
elements.  It  is  therefore  necessary,  in  the  absence  of  a  knowu 
cause  for  such  deviations,  to  derive  the  masses  of  the  i)lauets 
in<lependeutly  of  the  secular  variations.  In  the  case  of  Mars 
the  mass  is  obtained  with  all  necessary  precision  from  the  sat- 
ellites. It  is,  however,  different  in  the  case  of  Mercury  and 
Venus.  Here  no  resource  is  left  us  but  to  determine  them 
from  the  periodic  inecpialities.  As  the  inequality  produced  by 
Venus  in  the  Earth's  longitude  is  rarely  more  than  eight  sec- 
onds, it  might  seem  that  the  coetllcient  would  be  too  small  to 
obtain  a  sufficiently  precise  value  of  the  mass.  But  in  the 
case  of  observations  upon  the  Sun,  Mercury,  and  Mars  the 
error  of  the  determination  of  the  mass  in  question  may  be 
almost  indefinitely  reduced  by  multiplication  and  extension 
of  the  observations  without  danger  of  systematic  error. 

To  illustrate  this,  let  us  suppose  the  Sun's  longitude  to  be 
determined  with  a  meridian  instrument  only  once  a  year,  say 
at  equal  intervals  of  three  hundred  and  sixty-five  days.  Let 
the  longitudes  thus  observed  be  compared  with  an  ephemeris 
in  which  the  elements  are  affected  with  only  slight  errors. 
Leaving  out  of  consideration  the  periodic  perturbations  pro- 
duced by  the  planets,  the  comparison  of  the  observed  longi- 
tudes with  the  tabular  ones  through  an  entire  century  should 
be  nearly  constant.  Any  error  atfecting  all  the  longitudes 
alike  would  appear  as  a  coustant.    The  errors  of  mean  motion 


V2 


GENBUAL  OUTLINE. 


11 


IN 


wotihl  vary  unitbrinly  with  the  tiiiu'.  TIiuh  the  other  chMncnts 
woiihl  bo  nearly  coiiHtaiit,  and  couhl  be  Htill  more  upproxi- 
uiatcly  repreHeiited  by  a  Hli^lit  apparent  secuhir  variation. 

Now  let  the  diHturbin^  a(^tion  of  a  planet,  say  Venu»,  be  in- 
troduced. We  Hhould  tlien  have  a  series  of  deviations  from  the 
law  of  uniform  in(;rease,  whieh  would  enable  us  to  evaluate 
the  mass  of  the  planet.  The  value  of  this  mass  thus  derived 
would  not  be  aflected  by  any  systematic  error  common  to  all . 
the  observations,  nor  even  by  such  an  error  which  varied  uni- 
formly with  the  time.  Nor  would  small  errors  in  the  adopted 
elements  of  the  Sun  have  any  ett'ect  upon  the  result. 

If  this  would  be  the  case  for  observations  nnide>  only  at  a 
certain  point  of  the  orbit,  a  fortiori  w«mld  it  be  the  case  for 
the  observations  made  at  various  jmints  of  the  orbit,  since  any 
tendency  to  a  systematic  ettect  of  the  errors  of  observation 
W(Mild  thereby  be  ultimately  eliminated. 

Considerations  almost  identical  apply  to  the  case  of  observa- 
tions upon  either  of  the  planets  when  we  consider  the  action 
of  the  other  planet  upon  the  planet  observed  and  upon  the 
earth.  Hut  they  do  not  apply  to  the  case  of  the  action  of  the 
eaith  itself  Jipon  the  obser\ed  planet,  or  rice  rerna.  For  ex- 
am])le,  in  the  case  of  observations  of  Venus,  we  may  suppose 
that  all  observations  made  when  Venus  is  at  a  certain  point 
of  its  relative  orbit,  near  inferior  conjunction,  say  one  month 
before  inferior  conjunction,  are  affected  with  a  certain  error 
common  to  all  observations  made  at  that  point  of  the  orbit. 
Since  the  perturbations  produced  by  the  third  planet  will  in 
the  lon}>'  run  have  all  values,  positive  and  negative,  for  these 
several  observations,  the  systematic  error  in  question  will  not 
attect  the  ultimate  value  of  its  mass.  But  the  perturbations 
of  Venus  produced  by  the  Earth,  as  well  as  those  of  the  Earth 
produced  by  Venus,  will  not  have  all  values  in  such  a  case,  but 
only  special  ones  dependent  on  the  relative  position.  Hence, 
detfTminations  of  these  masses  might  be  aft'ected  by  errors  of 
the  kind  in  question.  We  conclude,  therefore,  that  the  mass 
of  the  Earth  can  not  be  satisfactorily  determined  by  the  peri- 
odic perturbations  which  it  produces  in  the  motion  of  any 
planet,  nor  that  of  Venus  by  observations  on  Venus  through 
its  periodic  perturbations  of  the  Earth. 


8] 


THANSITH   OF   VENUS   AND   MEUCUKY. 


13 


In  tho  solution  of  tlu>  equiitions  ot  condition  the  method  of 
least  srinares  Iia8  been  nsed  tlirou^^hont,  the  iirninf^enient  of 
th«'  work,  the  choice  of  (|uantitie8  to  bu  coiTcctc<l,  and  the 
accuracy  of  tiie  coctticicntH  bcin^  ho  choHon  as  to  uiininii/e  tliu 
fH'i'ixt  nieclumical  hibor  of  nuikin|<^  the  necessary  nniltiplica- 
tions.  The  adoption  of  this  method  was  necessary  in  order  to 
separate,  so  far  as  possibU>,  the  various  unknown  (piantitics 
anil  show  to  wliat  extent  their  vahies  wore  interdependent. 
Hy  no  otlier  method  of  eombiinition  couhl  so  hir};:e  a  number 
of  unknown  quantities  have  been  separately  determined  in  a 
way  whicli  would  have  been  at  all  satisfactory.  On  the  other 
hand,  in  combining-  the  tinal  results  and  deci«lin{^  upon  tlie 
values  of  the  i'orrections  to  be  adopte«l,  the  method  has  not 
always  been  applied,  for  reasons  which  will  be  deveh»ped  iu 
Chapter  IV. 

Introduction,  of  results  of  olm-rmtitniH  on  transitH  of  Vcnm  and 

Mercury. 

8.  In  the  ease  of  Mercury  and  Venus  the  observed  transits 
over  the  Sun  give  relations  between  the  corrections  to  the 
elements  more  accurate  than  those  ordinarily  derivable  from 
meridian  observations.  This  is  especially  the  case  with  Venus. 
The  value  of  these  observations  is  greatly  increased  by  the 
fact  that  they  are  made  when  the  planet  is  near  inferior  con- 
junction, and  therefore  nearest  to  the  Earth,  and  in  a  i)oint  of 
the  relative  orbit  where  meridian  observations  are  necessarily 
most  uncertain.  In  the  case  of  Venus  the  error  of  the  helio- 
centric place  will  be  more  than  doubled  in  the  case  of  the  geo- 
centric place  during  a  transit.  As,  however,  the  observation 
of  a  transit  gives  no  one  element,  but  only  an  equation  of  con- 
dition between  the  values  of  all  the  elements  at  the  epoch,  tlie 
only  way  of  treating  it  is  to  introduce  the  result  as  such  an 
equation,  with  its  appropriate  weight.  The  determination  of 
the  proper  weight  is  a  difficult  matter.  The  systematic  errors 
of  meridian  observations  are  such  that  the  theoretical  value 
of  the  weights  assignable  to  so  great  a  mass  as  we  have  dis- 
cussed would  be  entirely  illusory.  In  fact  so  great  is  the 
weight  assignable  to  the  observed  transits  of  Venus  that  if 
we  should  regard  the  results  of  each  transit  as  a  condition  to 


14 


GENERAL  OUTLINE. 


[8 


be  absolutely  natisfied  we  should  not  be  dangerously  in  error. 
I  conclude,  therefore,  that  there  is  more  danger  of  assigning- 
too  small  than  too  great  a  weight  to  these  observations. 

In  order  to  determine  what  change  was  produced  in  tlie  re- 
sults by  the  use  of  the  observed  transits  over  the  sun's  disk, 
two  sei>arate  solutions  of  the  equations  of  condition  for  Mer- 
cury and  Venus  were  made.  In  the  one,  termed  solution  A, 
the  meridian  observations  alone  were  used;  in  the  other, 
termed  solution  B,  the  combined  equations  formed  by  adding 
the  normal  equations  derived  from  the  transits  to  those  given 
by  the  meridian  observations  were  used. 

In  the  case  of  solution  A  it  was  originally  supposed  that  by 
using  the  mean  epoch  of  all  the  observing  in  the  case  of  each 
planet  as  that  from  which  the  time  was  to  be  reckoned,  the 
normal  equations  for  the  secular  variations  would  be  almost 
completely  separated  from  those  for  the  corrections  to  the 
elements  themselves.  The  separation  would  be  complete  were 
the  observations  at  different  epochs  similarly  distributed 
around  the  orbit.  But,  as  a  matter  of  fact,  it  was  found  that 
the  accidental  deviations  from  this  symmetry  were  so  couside" 
able  that  the  separation  could  not  be  regarded  as  complete. 
The  solution  was  therefore  made  by  successive  approximations, 
the  terms  depending  on  the  secular  variations  being  in  the 
first  approximation  dropped  from  the  normal  equations  for  the 
corrections  to  the  elements,  and  aftei  wards  included  when 
approximately  determined,  and  vice  versa. 

In  the  case  of  solution  B,  in  Avhich  the  transits  were  included, 
such  a  separation  did  not  occur,  and  the  equations  were  solved 
in  the  usual  rigorous  way  for  all  the  unknown  quantities. 


CHAPTER  II. 

DISCUSSION    AND    RESULTS    OF    OBSERVATIONS  OF    THE 

SUN. 

Treatment  of  the  liUjht  Ascensions. 

9.  The  meridian  observations  of  the  Sun  have  been  treated 
on  a  system  ditt'erent  in  some  points  from  that  adopted  in  the 
case  of  the  planets.  It  was  possible  to  simplify  the  treatment 
by  8upposi:ig  that  the  small  latitude  of  the  Sun  was  always  a 
definitely  known  quantity,  so  that  when  the  observations  were 
corrected  for  it  the  apparent  motion  of  the  Sun  could  be  sup 
posed  to  take  place  along  the  great  circle  of  the  ecliptic.  This 
allowed  the  correction  of  the  elements  to  depend  on  but  two 
quantities— the  obliquity  of  the  ecliptic  and  the  Sun's  true 
longitude.  Assuming  the  obliquity  to  be  known,  the  longi- 
tude of  the  Sun  could  always  be  determined  IVom  an  observa- 
tion of  its  Kight  Ascension.  An  observed  liiglit  Ascension 
being  compared  with  a  tabular  one,  the  residual  gives  rise  to 
an  ecpiation  of  C(mdition  between  the  correction  of  the  long- 
itude, A,  of  the  obliquity,  f,  and  of  the  Kight  Ascension  of  the 
Sun,  a\ 

da  =  cos  6  sec*  6dX  —  ^  tan  e  sin  2ad>;. 

Tiiis  equation  may  be  used  to  express  the  error  of  the  longi 
tilde  in  terms  of  the  error  of  tike  obliquity  and  of  the  Right 
Ascension  as  follows : 

6\  =  sec  e  cos*  66a  -f  ^  tan  f  sin  2Me 
=  s'^c  f  cos*  66a  +  0.21  sin  2\de 

The  elements  mainly  to  be  determined  from  the  observations 
in  Kight  Ascension  being  the  eccentricity  and  perihelion  of 
the  Earth's  orbit,  each  of  the  coefficients  of  which  go  through 
a  period  in  a  year,  the  effect  of  the  small  term  -  0.21  6e  sin  2\ 
whose  coefficient  does  not  amount  to  O'MO  after  1800,  and  has 
a  period  of  half  a  year,  will  be  practically  without  influence 

16 


16 


OBSERVATIONS  OF   THE  SUN. 


[10 


on  the  result.  The  system  was  therefore  adopted  of  deriving 
the  residual  in  longitude  directly  from  the  residual  in  Hight 
Ascension  by  the  formula 


where 


6\  =  Fda 
F  =  cos^  6  sec  e. 


h 


The  residual  6^  in  true  longitude  is  then  to  be  expressed  in 
terms  of  the  residual  61"  in  mean  longitude  and  of  corrections 
to  the  eccentricity  and  to  the  longitude  of  the  perigee  relative 
to  the  Stars.  In  this  expression  the  coefficient  of  the  residual 
in  mean  longitude  was  always  taken  as  unity,  the  value  of  the 
correction  being  so  small  in  the  case  of  Leverrier's  tables 
that  no  appreciable  error  would  result  from  this  supposition. 
Thus  each  residual  in  Right  Ascension  would  give  rise  to  an 
equation  of  condition  of  the  form — 

61"  +  Ve"67r"  +  E6e"  =  6X  =  ¥6a 

We  are  here  to  regard  61"  and  671"  as  corrections  to  the 
Kight  Ascensions  relative  to  the  clock  stars,  and  not  to  the 
Sun's  longitude  or  perigee  simply.  I  shall  therefore  use  the 
symbol  c  instead  of  61"  to  express  the  relative  correction  here- 
after. 

Treatment  of  the  Declinations. 

10.  The  declination  of  the  Sun  in  he  case  supposed  is  a 
function  only  of  the  longitude  and  0Dli(iuity.  The  equation 
♦or  exi)ressing  the  observed  correction  in  Declination  in  terms 
ot  *he  corrections  to  these  two  quantities  is 

/J6  =  sin  a6e  -|-  cos  a  sin  £6\ 

Thus  each  observation  of  the  Sun's  Declination  gives  rise  to 
an  equation  of  condition  of  this  form. 

It  is  however  to  be  supposed  that  the  observations  in  Decli- 
nation made  at  each  observatory  will  be  affected  by  a  constant 
error.  If  the  observations  are  truly  reduced  to  the  standard 
system  of  star  places,  this  error  will  be  that  of  the  standard 
system.  As  a  matter  of  fact,  however,  observations  made  in 
the  daytime,  especially  on  the  Sun  and  at  noon,  are  made 
under  circumstances  so  different  from  night  observations  on 


mimm 


llj       FORMATION  OF  EQUATIONS  IN  RIGHT  A£?CENSIONS.         17 

Stars  that  we  can  not  assume  the  error  of  the  reduced  declina- 
tion to  be  necessarily  the  same  as  that  of  the  star  system. 
We  must,  therefore,  in  each  ca^se,  regard  the  constant  error  in 
declination  as  something  peculiar  to  the  observatory  and  the 
instrument,  which  may  or  may  not  be  worthy  of  subsequent 
discussion.  Thus  each  residual  in  declination  gives  rise  to 
an  equation  of  condition, 

Jd„  +  cos  a  sin  eSX  +  sin  (xde  =  JfS 

/IS  being  the  excess  of  observed  over  tabular  declination, 
and  zl6„  the  common  error  of  all  the  measured  declinations  of 
any  one  series. 

ForhMtioii  of  the  equations  from  Rif/ht  Ascensions. 

11.  The  method  of  treating  the  observed  Kight  Ascensions 
of  the  Sun  was  suggested  by  the  fact  that  they  are  peculiarly 
liable  to  systematic  and  personal  errors;  tlie  former  likely  to 
change  with  the  seasons,  au«l  to  be  different  for  ditterent  in- 
struments ;  and  the  latter  to  continue  through  the  work  of  one 
observer.  It  is  now  well  understood  that  the  observed  Right 
Ascensions  of  the  mean  of  the  Sun's  two  limbs  relative  to  the 
fixed  stars  are  aifected  by  personal  errors,  no  means  of  elimi- 
nating which  have  yet  been  tried.  In  a  series  of  observations 
made  by  a  single  observer,  under  uniform  conditions,  this  error 
would  systematically  affect  only  tiie  relative  mean  of  the  Kight 
Ascensions  of  the  Sun  and  Stars,  leaving  the  eccentricity  and 
perigee  derived  from  the  observations  substantially  correct. 

On  taking  up  the  work  it  was  also  supposed  that,  owing  to 
the  different  effect  of  the  Sun's  rays  upon  the  instrument  at 
different  seasons,  and  the  different  circumstances  under  which 
observations  were  made,  the  Right  Ascensions  of  the  Sun 
would  berattected  by  errors  varying  in  a  regular  way  through 
the  year,  but  not  wholly  expressible  as  a  term  of  single  annual 
period.  It  was  therefore  deemed  best  to  consider  the  observa- 
tions possibly  affected  by  an  error  of  double  period,  having  the 
form 

x'  cos  2g  4-  y'  sin  2g 


5690  N  ALM- 


18 


OBSERVATIONS  OF  THE  SUN. 


[11 


The  introduction  of  the  coeiBcieuts  x'  and  y'  added  two  more 
terms  to  the  equations  of  condition,  which  terms,  however,  did 
not  express  any  astronomical  fact,  but  only  the  possible  errors 
of  the  observations. 

An  additional  and  very  important  element  to  be  determined 
from  the  observed  Right  Ascensions  was  the  mass  of  Venus. 
The  question  now  arose  whether,  by  a  uniform  series  of  cbser- 
vations,  extending  through  some  definite  period,  the  correc- 
tions to  the  eccentricity  and  perigee  and  the  coeflBcients  x'  and 
y'  could  br  completely  separated  from  the  coefficients  of  the 
correction  to  the  mass  of  Venus.  Examination  showed  that 
from  such  a  series  of  observations,  extending  through  eight 
years,  the  mass  of  Venv.s  could  be  determined  irrespective  of 
all  systematic  errors  repeating  themselves  with  the  season, 
provided  that  the  observc>,tion8  were  equally  distributed 
throughout  the  year,  or  even  that  an  equal  number  were  made 
at  the  same  time  through  successive  years.  As  neither  of 
these  conditions  are  practically  fulfilled  it  was  judged  best  to 
assume  in  the  beginning  that  the  systematic  errors  of  an  un- 
known kind  repeated  themselves  at  each  season  during  an 
eight-year  period,  and  that  they  could  be  expressed  in  the 
form 

c-\-  X  cos  g  -\-y  sin  g  +  x'  cos  2g  •{■  y'  sin  2g 

X  and  y  would  appear  as  errors  of  eccentricity  and  perigee 
which  could  not  be  eliminated. 

The  quantities  actually  introduced  as  the  unknown  ones  of 
the  equations  of  condition  were  as  follows: 

//',  the  factor  of  correction  of  the  mass  of  Venus ; 
a?,  one-fifth  the  correction  to  the  eccentricity; 
y,  one-fifth  the  correction  e"dn"\ 
x\  y',  one-tenth  the  coefficients  expressing  the  supposed 
error  of  double  period  arising  from  all  causes  whatever ; 

c,  the  constant  correction  to  the  Bight  Ascension  of  the 
Sun  relative  to  the  Stars. 

The  coefficient  of  c  was  supposed  unity  throughout.  The 
reduction  of  the  residual  in  Bight  Ascension  to  that  in  Longi- 
tude and  the  other  factors  were  taken  from  a  table  like  the 
following,  of  which  the  argument  was  the  day  of  the  year. 


4.^ 


v.^ 


Ill        FORMATION  OF  EQUATIONS  IN  BIGHT  ASCENSION. 


10 


Separate  tables  were  constructed  for  1802  and  ISfiO,  but  they 
were  so  nearly  identical  that  no  distinction  need  be  made 
between  them.  Furthermore,  the  error  introduced  by  sup- 
posing the  mean  anomaly  to  have  the  same  value  on  the  same 
day  of  every  year  is  entirely  unimportant. 

Table  of  coefficients  for  expressinfi  errors  of  the  Sun's  Right 
Ascension  in  terms  of  errors  of  the  elements  of  the  EartWs 
orbit. 


Coefficients  of — 

da 
dl 

dl 

da 

x=o.26e 

y=o.2e6n 

x' 

y 

Jan. 

I 

1.09 

0.  91 

+  0.1 

— 10.  0 

+  0.  I 

-f-io.  0 

II 

1.07 

0-93 

1.8 

9.8 

^•5 

9-4 

21 

1.04 

0.96 

3-4 

9.4 

6.5 

7.6 

31---- 

1. 01 

0.98 

S-o 

8.7 

8.7 

50 

Feb. 

lO 

0.98 

1. 01 

3.4 

7.7 

9.8 

+  1.8 

20 

0.96 

1.04 

-:-  7-6 

-6.S 

+  9-9 

—  1.6 

Mar. 

2 

0.94 

1.06 

8.6 

5- 1 

8.7 

4.9 

12 

0.  92 

1.08 

9.4 

3-5 

6.6 

7-S 

22 

0.92 

1.08 

9.8 

1.9 

3-7 

9-3 

Apr. 

I 

0.93 

1.07 

10. 0 

—  0.  I 

+  0.3 

10.  0 

II 

0.94 

1.05 

+  9-9 

+  1.6 

—  31 

-  9-S 

21 

0.  96 

1.03 

9.5 

3-2 

6.  I 

7-9 

May 

I 

0.99 

1. 01 

8.8 

4.8 

8.4 

5-4 

II 

1.02 

0.98 

7.8 

6.2 

9-7 

—  2.  2 

21 

1.05 

0-95 

6.6 

7-S 

9.9 

1.  2 

31--- 

1.07 

0.93 

+  5-3 

+  8.S 

-8.9 

-  4-S 

June 

lO 

1.09 

0.91 

3-7 

9-3 

6.9 

7-2 

20 

1. 10 

0.91 

2.  I 

9.8 

4.1 

9.1 

so- 

1.09 

0.91 

-1-0.4 

10. 0 

-  0.7 

10. 0 

July 

lo 

1.08 

0.93 

—  1-3 

9.9 

+  2.7 

9.6 

20 

I.  OS 

0.9s 

-  30 

+  9-5 

+  5-8 

—  8.2 

30-  — 

1.03 

0.97 

4.6 

8.9 

8.2 

5-7 

Aug, 

9-... 

1. 00 

1. 00 

6.1 

8.0 

9.6 

+  2.7 

19.-.. 

0.97 

1.03 

7-3 

6.8 

10. 0 

-  0.8 

29.... 

0-95 

I.  OS 

8.4 

5-4 

9.1 

4.1 

Sept. 

8.... 

0.93 

1.07 

-  9.2 

+  3-9 

+    7-2 

-6.9 

18.... 

0.  92 

1.08 

9-7 

2-3 

4-S 

8.9 

28.... 

0.92 

1.08 

10.  0 

-f  0.6 

+  1.2 

9-9 

Oct. 

8 

0.93 

1.07 

9.9 

—  I.  I 

—  2.2 

9-7 

18 

0.9S 

I.  OS 

9.6 

2.8 

5.4 

8.4 

28.... 

0.97 

!.02 

-  9.0 

—  4.4 

-  7-9 

-  6.1 

Nov. 

7 

I.  CO 

0.99 

8.1 

S-9 

9-5 

-  3-1 

17 

1.03 

0.96 

7.0 

7.2 

10.  0 

+  0.3 

27 

1.06 

0.94 

5.6 

8-3 

9-3 

3-7 

Dec. 

7 

1.08 

0.92 

4-1 

9.1 

7-S 

6.6 

17 

1.09 

0.91 

-    2-S 

-  9-7 

-  4.9 

-f  8.7 

27 

1.09 

0.91 

-  0.8 

— 10.0 

—  1.6 

+  99 

20 


OBSEEVATIONS  OF  THE  SUN. 


[12 


iH 


Finally,  throughout  the  work  the  equations  of  condition 
were  expressed  only  in  entire  numbers,  the  decimals  being 
neglected.  To  lessen  the  number  of  equations  of  condition, 
the  residuals  were  divided  into  groups  generally  covering  from 
ten  to  fifteen  days,  the  length  of  the  group  being  determined 
by  the  condition  that  the  perturbations  of  Venus  must  not 
change  nuich  during  the  period. 

While  the  formation  and  solution  of  the  equations  of  condi- 
tion on  this  system  were  going  on,  it  was  found  that  the  intro- 
duction of  the  assumed  coefficients  x'  and  y'  was  a  refinement 
productive  of  little  or  no  good  result.  In  fact,  the  observa- 
tions of  the  Sun  proved  to  be  much  freer  from  annual  sources 
of  error  than  I  had  supposed,  as  will  be  seen  by  the  tables  of 
their  results  soon  to  bei  given.  This  is  shown  by  the  general 
consistency  of  the  corrections  to  the  eccentricity  and  i)erigee 
given  by  the  work  at  the  same  or  diftercut  observatories  dur- 
ing dift'erent  periods. 

In  marked  contrast  to  this  is  the  discordance  among  values 
of  the  correction  c  to  the  relative  Right  Ascensions  of  the  Sun 
and  Stars.  This  quantity  it  is  that  is  affected  by  personal 
error  and  possibly  by  the  efiect  of  the  Sun  on  the  instrument. 
Under  a  perfect  system  of  discussion  it  would  be  advisable  to 
determine  it  separately  for  each  observer.  This  however  was 
practically  impossible. 

Solution  of  the  equations. 

12.  For  the  purposes  of  forming  and  solving  the  normal 
equations,  the  equations  of  condition  were  divided  into  groups 
of  generally  from  four  to  eight  years,  the  exact  lengths  of 
which  will  be  seen  from  the  following  exhibit  of  results.  The 
equations  for  each  period  were  solved  on  the  supposition  that 
the  corrections  were  constant  during  the  period.  Thus  every 
separate  result  is  independent  of  every -other,  except  so  far  as 
they  may  depend  on  the  same  instrument  or  the  same  observer 
at  different  times. 

The  first  column  shows  the  years  through  which  the  obser- 
vations extend. 

The  second  one  shows  to  the  nearest  year  the  value  of  T — 
that  is,  the  fraction  of  the  century  after  1850. 


121 


SOLUTION  OF  THE  EQi:ATIONS. 


21 


The  third  column  shows  the  value  of  /a',  or  that  factor  which, 
being  multiplied  by  the  adopted  mass  of  Venus,  is  to  be  applied 
as  a  correction  to  that  mass,  to  obtain  the  value  given  by  the 
observations. 

All  systematic  errors  arising  from  the  instrument  and  the 
observer  are  so  completely  eliminated  from  the  separate  de- 
terminations of  pi'  that  they  may  be  regarded  as  absolutely 
independent  of  each  other,  that  is — as  not  affected  by  any 
common  systematic  error. 

We  have  next  the  relative  weight  assigned  to  each  value 
of  yu',  which  is  determined  in  the  usual  way  from  the  Solu- 
tion, and  is,  therefore,  on  a  different  scale  for  different  ob- 
servatories. 

Next  is  given  the  value  of  c,  or  the  apparent  correction  to 
the  Right  Ascension  of  the  Sun,  relative  to  the  assumed  Ilight 
Ascensions  of  the  Stars,  as  given  by  observations  during  the 
several  periods  and  expressed  in  seconds  of  arc,  followed  by 
the  weights  assigned  to  the  separate  results. 

The  next  two  columns,  the  corrections  to  the  solar  eccen- 
tricity and  to  the  longitude  of  the  perigee,  require  no  further 
explanation. 

Respecting  the  weights  ultimately  assigned  to  these  quanti- 
ties, and  to  c,  it  is  to  be  remarked  that  they  are  the  result  of 
judgment  more  than  of  computation.  It  is  only  possible  to 
enumerate  in  a  general  way  with  some  examples  the  consider- 
ations on  which  they  are  based. 

In  assigning  the  weight  of  c  the  number  of  observers  en- 
gaged is  an  important  factor  in  determining  it.  Other  factors 
are  the  steadiness  of  the  atmosphere  and  the  adaptation  of  the 
instrument  to  this  particular  work.  General  consistency  is 
an  important  factor  in  the  assignment.  In  this  respect  the 
Cambridge  observations  are  quite  remarkable  ;  if  their  excel- 
lence corresponds  to  their  consistency  they  must  be  the  best 
ones  made. 

It  will  be  seen  that  Piazzi's  results  are  thrown  out  en- 
tirely. The  wide  range  of  his  values  of  c  led  to  the  inquiry 
whether  more  consistent  results  would  be  obtained  by  taking 
shorter  periods,  but  it  was  found  that  the  values  of  c  varied 
from  time  to  time  in  such  an  irregular  way  that  his  instrument 


22 


OBSERVATIONS  OF  THE   SUN. 


[12 


uiust  liave  beeu  att'ected  by  some  extraordiutiry  cause  of  error, 
unless  some  mistake  bus  beeu  made  hi  interpreting  or  treating 
tbe  observations. 

Tbe  Oxford  values  of  c  are  unusually  discordant.  Tbe  pre- 
sumption tbat  tbis  discordance  arises  mainly  from  tbe  special 
personal  e<iuation  in  observations  of  tbe  Sun,  described  on 
page  17,  derives  additional  w  sigbt  from  tbe  greater  relative 
consistency  of  tbe  values  of  dc"  and  e"6n".  I  bave  tberefore 
allowed  tbe  values  of  tbese  quantities  to  receive  a  fair  weigbt. 

Tbe  value  of  c  for  Paris,  1800-70,  lias  received  a  mncb  re- 
duced weigbt,  solely  on  account  of  its  excessive  value.  It 
seems  tliat  tbe  work  of  one  observer  wbo  made  many  observa- 
tions during  tbis  period  was  att'ected  by  an  unusual  system- 
atic error. 

Results  of  observations  of  the  Sufi's  Right  Ascension. 

GREENWICH. 


Years. 

r 

,"' 

re 

c 

w 

6ef' 

e'^iSn'^ 

70 

II 

II 

II 

l75o-'62 

-•94 

—.027 

20 

+0-33 

'•5 

-1-0.04 

— 0.42 

2 

1765-71 

— 

82 

—  .041 

10 

+0.37 

0-5 

—0.08 

—0.  64 

I 

i772-'78 

— - 

75 

—  .022 

10 

+0.74 

0-5 

—0.  16 

—0.49 

I 

I779-'8S 

— 

68 

-•035 

s 

+2.89 

0.  2 

—0.  18 

—0.  73 

0.5 

i786-'92 

— 

61 

-•037 

8 

+  1-5' 

0.  2 

— 0.  12 

—0.88 

0 

1793-97 

— 

55 

— .  114 

S 

+  1.87 

0.  2 

—0.  22 

-1.27 

0 

i798-'o2 

— 

50 

+  .060 

5 

-|-i.  02 

0.  2 

-  0.  42 

-  I.  IS 

0 

i8o3-'o6 

— 

45 

— .002 

5 

+0.27 

0.  2 

—0.03 

-1.03 

0 

i8o7-'io 

— 

41 

-.068 

5 

—0.  34' 

0.  2 

—0.32 

—  1. 12 

0 

i8ii-'i4 

— 

37 

—•095 

3 

-3-33 

0.2 

+0.17 

—  I  08 

0 

I8is-'i8 

— 

33 

—  .052 

6 

-1.99 

o.S 

— 0.  12 

—0.34 

0 

l8l9-'22 

— 

29 

-f  .010 

6 

-0.51 

-fo.  22 

—0. 19 

I 

1 823-' 26 

— 

25 

-.054 

6 

—  1.08 

* 

+0.05 

—0.17 

I 

i827-'30 

— 

21 

-■  045 

6 

—0. 42 

1 

— 0.  09 

-0.75 

I 

1 83 1 -'34 

— 

17 

+.016 

7 

+0.76 

+0.04 

—0.  27 

I 

1835-38 

— 

13 

-J-.020 

8 

+  1.  16 

-fo.  26 

4-0.06 

2 

i839-'42 

— 

09 

-i-.o6i 

8 

+0.84 

4-0.32 

-j-o.  10 

2 

i843-'46 

— 

05 

-."008 

8 

+0.15 

2 

+0-25 

-|-0.  22 

2 

i847-'5o 

— 

01 

—•045 

8 

—0.  10 

2 

4-0.28 

4-0.02 

3 

i85i-'S4 

+ 

03 

+.024 

8 

4-0.40 

3 

4-0.  22 

4-0.02 

3 

1855-58 

+ 

07 

-.  032 

9 

f  0.  36 

3 

+0.  IS 

4-0.  02 

3 

i859-'62 

+ 

II 

—.043 

9 

—0.02 

3 

4-0.25 

4-0.  22 

4 

i863-'66 

+ 

15 

—.016 

8 

+0-3I 

3 

4-0.23 

— 0.  05 

4 

i867-'7o 

+ 

>9 

+.031 

8 

+0.35 

3 

+0.33 

—0.  10 

4 

i87i-'74 

+ 

23 

+.02 1 

8 

-fo.  12 

3 

+0.24 

4-0.05 

4 

1875-78 

+ 

27 

—.008 

8 

— 0.  12 

3 

4-0.26 

4-0.06 

4 

i879-'82 

-f 

31 

+.017 

8 

—0.05 

3 

4-0.  21 

4-0.14 

4 

i883-'88 

+ 

36 

-f.  001 

«3 

—0.  20 

3 

4-0.18 

+0.07 

4 

i889-'92 

+ 

4« 

-.025 

8 

-0.44 

2 

4-0.24 

■ 

4-0.  II 

3 

12J  SOLUTION  OF  THE  EQUATIONS.  23 

KesultH  of  ohfiervations  of  the  (S'mm'»  h'iffht  Anfemion — Continued. 

PARIS. 


Years. 

T 
-.46 

-.025 

Ji' 
14 

t" 

to 

•_ 

1  0.08 

u> 

1801-07 

II 

-1.78 

i8o8-'is 

—  .^8 

+-o«5 

17 

—0.65 

0.5 

-  0.  01 

f  0.  12 

I 

l8l6-'22 

—  3> 

— .  050 

14 

+0.  18 

0.5 

-0.  13 

+0. 32 

i823-'29 

-.24 

—.050 

10 

-f-o.  01 

o.s 

-0.31 

—0.  02 

1837-44 

—.09 

-.034 

19 

+0.33 

I 

—0.  04 

+0.  10 

"■5 

i845-'52 

+.01 

+.009 

15 

-f  0.  10 

I 

+0.04 

-f  0.  10 

1-5 

>85.5-'59 

+  .06 

+.014 

15 

+0.66 

I 

— 0.  04 

+0.32 

2 

1 860- '6s 

+■13 

+■003 

10 

+0.38 

I 

4-0.07 

f  0.  26 

2 

i866-'7o 

-t-.i8 

.000 

7 

+2.29 

0.3 

+o^  »3 

4-0.  40 

2 

i87i-'79 

+  •25 

+•048 

II 

—0.  26 

I 

—  0.06 

4-0.  22 

2 

1 880-' 89 

+  ■35 

-f-.002 

14 

4-0.44 

I 

+0.24 

-4-0.03 

2 

PALERMO. 


i79i-'96 

—  56 

-.079 

0 

—0.07 

0 

II 

-  -0. 06 

—0.85 

0 

i797-'oi 

—  5' 

-.116 

0 

-2.33 

0 

—0.  29 

-0.  28 

0 

i8o2-'os 

-.46 

—.001 

0 

-3^" 

0 

—0.05 

^0.76 

0 

i8o6-'i2 

—.41 

+  ■243 

0 

-hs.  92 

0 

-1. 17 

+  •■55 

0 

cami5Rii)c;k. 


1 828-' 34 

—.21 

+  •007 

16 

-0.  13 

2 

-fo.  08 

4-0.  12 

4 

1835-40 

— .  12 

-•033 

14 

0.  18 

2 

-fO.06 

—0. 06 

4 

i8|2-'47 

-OS 

— .  026 

9 

—  0.  21 

2 

-fo.o8 

—0.  12 

4 

i85o-'s8 

+■04 

— .  024 

20 

—0.  U 

2 

4-0.17 

—0.04 

4 

WASHINGTON. 


i846-'52 
i86i-'6s 
1 866-' 73 
i874-'8i 
i882-'9i 


—  .01 

-.038 

5 

— o!'85 

2 

4-0.  20 

1 
0.00 

+■13 

-■  038 

8 

-0.53 

4 

-j-o  01 

0.00 

-^.  20 

— .  004 

J3 

—o.  22 

4 

+0.18 

—0.  03 

+.28 

-■033 

12 

—0.45 

4 

4-0.  07 

—0.  i6 

,+.37 

—.002 

17 

—0.79 

4 

+0.  07 

-0.07 

3 
S 
6 

5 
5 


KONIGSBERG. 


i8i6-'23 

—•30 

4-.C02 

»3 

4-0! '30 

I 

4-o'.'o7 

—0.28 

3 

i824-'^o 

-■23 

— .006 

12 

4-0.  02 

I 

~o.  16 

4-0.  II 

3 

1 83 1 -'38 

—  «S 

— .021 

«S 

4-0.23 

I 

— 0.  12 

+0.03 

3 

i839-'4S 

—.08 

— .021 

12 

4-0.77 

I 

+  0.08 

-f  0.  20 

3 

24  0B8EUVAT10NS  OF   THE  SIN.  [18 

Results  of  ohHerrationtt  of  the  Sun'ti  Ri<jhi  Ascemion — Cuutiimed. 

OXFORD. 


Years. 

T 

-•OS 

+.14 

+■23 

+  ■34 

// 

w 

12 

«3 
•S 

9 

*■ 

w 

n 

+0.24 
+0.08 
+0. 20 
+0.27 

II 
-0.17 

-0.13 

—0.04 

+0.64 

w 

i840-'49 
i860-' 68 
1 869-' 76 
i88o-'87 

—■043 
+  .042 

+  •054 
—  .014 

// 

+2.49 
+  1.96 
+0.92 
-0.31 

0.3 
0.3 
0.3 
0.3 

2 
2 
2 

2 

I'ULKOWA. 


i842-'so     —.04      +.047 
i86i-'7o     +.16     4-002 


1 1      -f  1 .  20       I 
10  1  —0.40       I 


" 

//                  1 

-0. 

12 

+  0. 

20 

+0. 

05 

+0. 

28 

DOR  PAT. 


It 

// 

// 

1 823-' 30 

-•23 

-t-.02I 

9     +©■  36 

I 

—0.  12 

—0.  22 

i83i-'38 

-■IS 

+  .008 

6 

+0.45 

I 

+0.02 

+0.03 

CAPE  OF  GOOD  HOPE. 


i884-'90 

+•37 

— .026 

12      -o."36 

3 

II 
4-0.02 

+0.01 

4 

STRASSBURG. 

i883-'88 

+.36 

—.014 

"    '  1 
12      —1. 6s  1     2 

+0.23 

+o'.'o9 

3 

The  mass  of  Vemis. 

13.  The  mean  results  for  the  mass  of  Venus  given  by  the 
work  at  the  several  observatories  are  shown  as  follows: 

The  probable  error,  where  given  at  all,  is  that  derived  from 
the  discordance  of  the  separate  individual  results  at  the  par- 
ticular observatory.  In  some  cases  there  are  only  one  or  two 
results ;  here  no  probable  error  could  be  assigned. 

w'  is  the  sum  of  the  weights  of  the  result  at  each  separate 
observatory,  as  given  by  the  equations  of  condition.  Were 
all  the  observations  of  equal  accuracy,  these  would  be  the 
weights  to  be  assigned  to  the  separate  lesults.    Such  not  be- 


141 


CORRECTIONS  OF  RELATIVE  RIGHT  ASCENSIONS. 


25 


ing  the  case,  we  choose  for  the  actual  weights  certain  numbers, 
founded  partly  on  a  compromise  between  the  mean  errors  fol- 
lowing each  result  or  upon  the  values  of  w',  partly  on  a  judg- 
ment of  the  accuracy  of  the  observations. 

t'aluea  of  jit'  /or  the  mass  of  f'eiiua. 


Greenwich 

I'aris 

Kdnigsberg 
Cambridge 

Dorpat. 

Pulkowa  .. 

Oxford 

Washington 

Cape 

Strassburg . 


/*' 

W 

IV 

-.oisd 

-.006 

226 

II 

—.007- 

-.<X)9 

146 

5 

—.012- 

-.010 

52 

3 

-.018- 

;,  009 

59 

6 

-f-.oi6 

'5 

I 

+  .025 

21 

I 

+  .oi4-{-.023 

49 

1 

— .  oiS-i-.  009 

SS 

4 

—.026 

12 

I 

—.014 

12 

I 

Fsing  the  weights  in  the  last  column,  we  have  for  the  mean 
result 

/ii=  -  .0118  ±  .0034. 

The  mean  error  i  .0034  is  that  given  by  the  discordance  of 
the  separate  results  of  the  preceding  table. 

Corrections  0/ relative  Bijfht  Ascensions. 

14.  The  true  values  of  the  remaining  quantities  c,  Se",  and 
€"67t"  are  to  be  regarded  as  increasing  uniformly  with  the 
time  and  therefore  of  the  form 

x  +  Ty. 

Here  T  is  the  time,  and  in  the  treatment  of  these  particular 
equations  it  is  counted  from  1850  in  units  of  one  century,  so 
that  X  is  the  value  of  the  correction  at  this  mean  epoch. 

The  quantity  designated  by  c  is  the  same  which,  elsewhere 
in  this  discussion,  is  represented  by  d'l  +  a,  so  that 

e  =  61"  4-  a 

I  shall,  however,  for  convenience,  continue  to  use  the  designa- 
tion c,  or  aj+Ty. 


,1 

^1 


20 


onSEKVATIONS  OF  THE  Ml'N. 


[U 


As  th(i  obaervrttions  at  (Ireeinvich  and  I'jiris  extend  over 
lon;;(M'  periods  than  at  any  other  obnervatorieH,  [  .sliall  first 
solve  them  separately.  The  totality  of  the  (Ireenwiidi  obser- 
vations  give  for  v  the  following  normal  eqnations  and  solution: 

43.4  A  +  1.0.")y=  +  V'.2:\ 
IM  +  4.24     =  -  1".25 

JO  =+i)".ll 
\j=  -  iV'M 

Those  at  Paris  give  the  eipiations  and  solution 

8.a.r-f  0.04y=  +  1".22 
0.04  4-  0.48    =  +  0".77 

a,- =4-0".  14 
y  =  +  1".59 

If  we  combine  all  the  other  results  into  a  single  set  of  normal 
equations,  we  have 

40.2.»  +  4.20y=-10".84 
4.20  +  2.20    =  -    3".08 

J^-s:  -0".10 

y=-  1".(I2 

It  will  be  seen  that  the  results  for  y,  the  secular  motion,  are 
markedly  discordant.  Indeed,  if  we  refer  to  the  exhibit  of 
results,  p.  23,  we  shall  see  that  the  values  of  c  are  nuich  more 
discordant  than  those  of  the  other  two  quantities.  To  obtain 
a  definite  value,  founded  on  all  the  observations  of  the  Sun's 
Right  Ascension,  I  do  not  see  that  any  bettor  result  can  be 
obtained  than  that  found  from  a  general  solution  of  the  com- 
bined normal  equations.  The  tifjuations  and  their  solution  are 
as  follows : 

91.0d7+5.0.5y=-o".39 
6.05  +  0.02    =  -  4".46 

x  =  -  0".02 
y=-  0".63 


or 


61"  +  «  =  -  0".02  -  0".63T 


15]     CORR.  TO  THE  SOLAR  ECCENTRICITY  AND  PKItlOKE.       27 

Cotrcctionn  to  the  Molar  evcvntririty and  pniyve. 

la.  I  have  iilroinly  inentioiu'd  tlie  reiimrkabU)  cronMisteiiry 
of  the  corn^ctioiis  to  rlieso  elenuMits  Kiveii  by  the  results  at 
ilitlrreiit  observatories  and  at  dirtereiit  epochs.  The  eceen- 
tiieity  Ih  more  consistent  than  tlie  perijice.  One  cause  foi' 
this,  the  consideration  of  which  will  throw  some  li^ht  on  the 
rehitive  merits  of  the  observations,  iH  that  the  error  of  Kight 
Asc«'nsion  depending  on  the  Declination  of  the  object  observed 
effects  the  eccentri<'ity  less  than  the  perijjree.  It  is  well  known, 
from  ii  c<»mparison  of  the  results,  that  the  systematic  ditVer- 
ences  in  the  Kifjht  Ascensions  of  different  star  catalo^iues 
vary  somewhat  with  the  Declination.  Now,  since  the  8un's 
Declination  goes  through  an  annual  period,  it  foll<)Ws  that  this 
error  will  produce  a  systenuitic  eflect  on  both  the  eccentricity 
and  the  perigee.  Hut  the  effect  will  be  much  larger  in  the 
case  of  the  latter  element  than  in  the  case  of  the  former, 
because  of  the  nearness  of  the  perigee  to  the  winter  solstice, 
the  difference  being  only  some  10°  or  IL"^.  (Jonse^piently  the 
extreme  coeflicients  in  the  correction  to  the  eccentricity  have 
nearly  the  same  values,  with  opposite  signa,  for  the  same  1  )ecli- 
nations  in  different  seasons  of  the  year.  lUit  it  is  different 
with  the  perigee.  The  coeflicient  of  this  quantity  is  negative 
from  October  until  March,  when  the  Sun  is  in  south  Declina- 
tion, attaining  its  maximum  value  about  Jannary  I ;  while  it 
IS  positive  during  the  remaining  months  when  the  tSun's  Decli- 
nation is  north,  attaining  its  maximum  value  about  July  1. 
A  systematic  difference  in  the  errors  of  liight  Ascension  will 
therefore  produce  its  full  effect  on  the  longitude  of  the  perigee, 
while  its  effect  on  the  eccentricity  will  be  but  slight. 

In  this  (ionnection,  the  very  large  negative  values  of  the  cor- 
rection to  the  perigee  during  the  period  when  the  old  (Ireen- 
wich  transit  instrument  was  in  use  are  ipiite  remarkable. 
The  progressive  change  in  the  value  of  c  is  also  remarkable  in 
this  connection.  It  is  to  be  remarked  that  the  new  transit  was 
mounted  in  181(),  but  account  was  not  taken  of  this  fact  in 
grouping  the  eciuations.  Hence  it  is  only  from  the  year  1819 
that  the  results  of  the  table  are  derived  wholly  from  observa- 
tions with  the  new  instrument.    The  anomaly  alluded  to  is 


•28 


OBSERVATIONS  OF  THE  SUN. 


[15 


then  seen  to  disappear.  The  fact  that  the  abnormally  large 
corrections  in  c  are  positive  before  1800  and  negative  after  it, 
while  e"6n"  is  abnormally  negative  through  the  doubtful 
period  1765-181"),  complicates  the  theorj  of  these  errors.  I 
have  not  been  able  to  consider  them  in  detail,  but  have  simply 
rejected  the  results  for  Se"  and  e"  dn"  from  1786  to  1818,  hav- 
ing given  them  a  gradually  diminishing  weight  from  Brad- 
ley's observations  to  tlie  first  epoch. 

As  in  the  case  of  c,  I  have  made  a  solution  for  Greenwich 
alone,  Paris  alone,  the  other  observatories  combined,  and  all 
combined.    The  results  are  shown  as  follows: 

i.  From  Greenwich  observations : 

8e"  e"Sn" 

54.5a'  +  2.73  J/  =  +  11".14;  -  0".88 

2.73  +  5.72    =  -[-    1".82;  +  2".m 

x=+    0".19;  -0".04 

y=+    0".22;  +  0".49 

2.  From  Paris  observations : 

Se"  e"87t" 

n.Ox+0:,V,)y=  +  0".30;  4-  2".95 

0.39  +  0.99    =  -f  0".29;  +  0".33 

^=+0".01; +0M7 

i/=+0".29; +0^27 

3.  The  equations  an<l  results  from  all  the  other  modern 
observations  are — 


8e" 


e"8n' 


77.00?+  4.992/  =  +  5".58;  +  0".35 

4.99  +  3.68    =  +  1".09;  +  0".40 

07  =  +  0".06;       0".00 

y=  +  0".22;  +  0".05 


lOJ     KESULTS  OF  OBSERVED  DECLINATIONS  OF  THE  SUN.      20 

4.  Finally,  if  we  combine  all  the  equations,  we  iiave — 

Se"  e"67r" 

148.5  J-  +    8.11  y  =  -I-  17".0L>;  +  2"A2 

8.1    -4-10.;3i)    =+    3".20;  +3''.42 

jp=-\-    (V'.IO;       o".m 

y=+    0".23;  +  0".33 

In  the  case  of  the  ecceutricity  the  gv°neral  accordance  is 
quite  p  itisftictory,  and  for  the  perigee  it  is  much  better  than 
in  the  case  f,  the  relative  liight  Ascension. 

Kestilts  of  observed  ileclinationa  of  the  Sun. 

10.  The  Sun's  absolute  longitude  can  be  found  <mly  from 
observations  of  his  declination,  because  this  longitude  is 
leferred  to  the  equinox,  which  is  delinetl  only  by  the  Sun's 
crossing  of  the  equator. 

The  corrections  to  the  eccentricity  and  perigee,  asjust  found, 
are  so  slight  that  they  may  be  neglected  in  determining  the 
correction  of  the  absolute  longitude  f'.cm  that  of  the  declina- 
tion. Thus,  as  already  stated,  the  unkrown  (juantities  of  tiie 
equations  given  by  the  declinations  are  the  corrections  of  the 
mean  longitude  I",  and  of  the  obliquity  f,  and  a  constant  Jrf, 
peculiar  to  each  observatory,  of  which  we  take  no  further 
account.  The  equation  of  condition  given  by  each  observa- 
tion or  group  of  observations  is 

J<S  +  A  sin  edl"  +  B')e  =  do 

where  dd  is  the  excess  of  the  observed  over  the  tabular  decli- 
nation, and 


d6 
A  =  cosec  e-TT  =  cos  a 
dx 


B  = 


ds 

d6 


=  sma 


!:! 


30 


OBSEFVATIONS  OF  THE  SUN. 


[16 


The  equations  are  grouped  and  solved  for  periods,  as  in  the 
case  of  the  iiight  Ascensions,  with  the  results  shown  in  the 
following  table: 

Results  of  rbservations  of  the  Snri's  Declination. 

GREENWICH. 


Years. 

T 

f!/'^ 

70 

de 

7V 

J<J 

,5^. 

70 

// 

// 

// 

// 

« 753-57 

-•95 

+0.78 

—0.  34 

I 

-2.43 

—0.34 

I 

i758-'62 

—.90 

+  I-50 

-I.  81 

I 

—1.94 

—  1. 81 

I 

i765-'70 

—.82 

—0.23 

0. 95 

OS 

-f  0.  20 

—0.95 

0.5 

1771-78 

-.75 

+0.  48 

—0.93 

©•5 

+  '•25 

—0.93 

©•5 

i779-'85 

—.68 

+  123 

— 1.09 

o^S 

—0.99 

— 1.09 

05 

i786-'9i 

—.61 

+0.48 

— 0.  50 

0.3 

+0.  IS 

— 0.  50 

03 

1792-97 

-•55 

-fi.  12 

—0.  70 

0,  2 

— 0.  35 

— 0.  70 

0.  2 

1798-03 

—•49 

-fo.41 

— 1.02 

0. 1 

— 0. 10 

— 1.02 

0. 1 

i8o4-'io 

-•43 

4-0.18 

—  1.41 

0. 1 

—0.84 

—1. 41 

0. 1 

i8i2-'i6 

—  36 

-0.15 

3 

-o^  53 

3 

+0.48 

-0.53 

3 

l8l7-'22 

-•30 

-0.  41 

3 

-f  0. 03 

3 

-\-o.  40 

+0.03 

J 

i823-'28 

--.  24 

+0.43 

3 

—  0.  10 

3 

-fo.o8 

— 0.  10 

•5. 

1829- '34 

—.18 

—0.08 

7 

-f  0.  21 

3 

+0.25 

+0.21 

3 

i83S-'40 

— .  12 

-0. 12 

3 

— 0.  20 

3 

+0.37 

-0.13 

3 

1 84 1 -'46 

— .   6 

+0.21 

3 

+0.13 

3 

+0.47 

-fo.  11 

4 

i?47-'52 

0 

+0.25 

4 

0.  CX) 

— 0.  24 

— 0. '5 

4 

1853-58 

+  •    6 

+0.55 

5 

+0.18 

—0.  26 

-0. 05 

5 

i859-'64 

+•12 

+0.03 

5 

+0.28 

—0.46 

+0. 12 

5 

i86s-'70 

+.i8 

—0.23 

5 

-0.15 

+0.05 

—0.36 

5 

i87i-'76 

+•24 

-0.15 

5 

-\-o.  26 

+0.  16 

— 0. 16 

5 

i877-'82 

+•30 

— 0.  90 

5 

4-0.  22 

-f-o.  34 

+0.08 

5 

i883-'88 

+•36 

—0.  27 

5 

+0.33 

—0.  14 

+0.02 

5 

i889-'92 

+.41 

—0  05 

3 

-j-o.  19 

,    3 

+0.13 

— 0.07 

3 

PARIS. 


i8oo-'o3 
i8o4-'o7 
i8o8-'io 
i8ii-'i5     — 
i8i6-'2i     — 

l822-'28      — 

l837-'42     — 
1843-48 

1849-54 
i8s5-'bo 
i86i-'66 
l867-'72 

!  873-' 7  7 
i878-'83 
1 88^ -'89 




48 

— 

44 

— . 

41 

— 

37 

— 

31 

— 

25 

— 

10 

— 

4 

+ 

2 

+• 

8 

+ 

14 

+ 

20 

+ 

25 

+ 

31 

+ 

37 

4-0. 01 

+0.70 

-1-2.66 
— o.  92 

+0.58 
+1.09 

fo.  79 

+0.43 
+  1.19 

+0.35 
+  1-35 
+0.31 
-0.59 
—  0.09 
—0,80 


-1-93 
-1-0,82 

-I-1.60 
—  1 .  20 

-I-1.68 

3  J  +o-  39 
-0.15 
—0.03 
— o.  01 
— o.  02 
0.00 
— o.  67 
-f  o.  04 
—0.32 

+0  32 


—0.45 

— 2.02 

—0.95 
—1. 18 

—1.42 

— o.  01 

4-0.40 
4-0.19 

4-»-34 
-\-\.  22 
4-0.  12 
4-0.  10 

-j-I.OI 

+0.58 
+0.78 


16]     RESULTS  OF  OBSERVED  DECLINATIONS  OF  THE  SUN.       31 

Results  of  observations  of  the  Sun^s  Declination — Continued. 

PALERMO. 


Years. 

T 

(5/'' 

W 

dt 

vv 

Jr5 

(!'f 

W 

1791-03 
i8o4-'ij 

-■53 
—.41 

// 
—  1.46 
-hi.  70 

0 
0 

// 

— 0-95 
—0.  52 

// 

4-0.78 
-1-0.42 

—0.95 
-0.52 

0.4 
0.4 

CAMBRIDGE. 


1833-38 
i839-'44 

1847-53 
1854-' 58 


— .  14 

— 0.  21 

2 

-0-33 

+0.  59  i 

—.08 

+0.31 

2 

— 0.  20 

-1-0.29  1 

00 

4-0.  21 

2 

+0.31 

—0.  ^2 

-f.o6 

-0..5 

' 

+0.34 



—0.42 

-0.  54 
— o.  41 
4-0.  10 
+0.  13 


WASHINGTON. 


i846-'49 
!S6i-'66 
i867-'72 

i873-'78 
i879-'84 
i885-'9i 


KOMGSBERG. 


1*7  <■/ 


1815 

l820-'23 

i824-'27 
i828-'3i 
i832-*34 
1837-44 


•35 
.28 
,24 
.  20 

1/ 

,09 


— o.  14 
+0.65 
4-1.08 
— o.  72 

—0.66 


— O.  22 
4-0.49 
-1-0.09 
-0.15 
— O.  02 


-0.59 

— o.  60 

-  O.  64 

—1.32 

— 2.  24 


-  I 

07 

0 

— 0 

47 

+0 

24 

— 0. 

16 

— 0 

40 

' 

— 0. 

87 

OXFORD. 


i840-'45 
i846-'5i 
i86i-'66 
i867-'72 
i873-'76 
i8So-'83 
i884-'87 


-.07 
— .01 

-H.  14 
4-.  20 

+  ■25 

+■32 
4-36 


4-0.79 
+0.35 
4-0.36 
— o.  16 
— c.  38 

— 0-43 
— o.  24 


4-0.42 
+0.40 
—0.81 
— o.  24 
-o.  33 
4-0.  12 
4-0.23 


4-0.67 
4-0.89 
4-0. 10 
4-0.29 
4-0.29 
— o.  17 
).  19 


4-0.  22 
j-o.  20 
—  1. 01 
—0.44 

-o  53 
— o  08 
4-0.03 


O.  2 
O.  2 
O.  2 
O  2 
O.  2 
O.  2 
O.  2 


i\ 


i 


32  OBSERVATIONS   OF  THE   SUN.  [16,  17 

Results  of  observations  of  the  Snn^s  Declination — Gontinued.  ' 

PULKOWA. 


Years. 

T 

f>l" 

W           6e 

1 

w 

A6 

&'t 

\V 

l842-'4S 
l846-'49 
i86i-'6'; 
i866-'7o 

-.06 
—.02 

+  •13 
+  .18 

// 

+0.82 
—0. 10 

-'>-53 
+0.27 

2 
2 

2 
2 

// 

-0.35 
—0.48 
—0.48 
-0.31 

// 

— 0. 01 
-fo.07 
—0.30 
-0.38 

// 

—0.  35 
—0.48 
—0.48 
—0.31 

I 
I 
I 
I 

DORPAT. 

l823-'28 
i829-'32 
•833-'38 

-.24 
-.19 
—.14 

+0.99 
+0.99 
+  1.00 

2 
2 

2 

-1.26 
—0.  76 
-0.63 

+0-59 

+1-34 

- +1-34 

—  I. 41 
—0.91 
—0.78 

1 
I 
I 

CAPE  OF  GOOD  HOPE. 

i884-'87 
i888-'90 

+•36 
+•39 

—0.51 
—0.84 

4     +0. 05 
4     +0.09 

+0.  II 
+0.19 

— 0.  07 
— 0.  21 

2 
2 

STRASBURG. 

i884-'88 

-!-•  36 

-0.57 

4 

—0.05 

—0.77 

+0.  12 

2 

leidi:n. 

» 

1 86/  -'69 
i87o-'76 

+■17 
+•23 

+0.14 
—0.  23 

4 
4 

— 0.  01 
—0.06 

+0.27 

— 0. 04 

— 0.  24 
—0.  29 

2 

2 

Correction  to  the  Sun''s  absolute  longitude 

17.  So  far  as  mere  instrumental  measurement  is  concerned, 
the  correction  6  e  should  be  determined  with  greater  precision 
than  61"  in  the  ratio  5:2,  because  the  errors  in  declination 
have  to  be  divided  by  the  factor  sin  f  =  0.40,  in  order  to  form 
61".  Mlowing  for  this  large  increase  in  the  source  of  error, 
the  values  of  61"  are  more  accordant  than  those  of  6€.  This 
is  what  we  should  expect.  The  values  of  the  former  quantity 
depend  mainly  upon  the  comparison  of  observations  made 


OBLIQUITY  OF  ECLIPTIC. 


33 


W 



!.■) 

I 

8 

I 

8 

I 

I 

I 

I 

I 

r 

I 

{ 

I 

2 
2 


2 

2 


17,  18] 

near  the  opposite  e(iuiiioxes,  when  the  sun  has  the  sanie  decli- 
nation, and  wlien  the  season  is  not  greatly  different.  Indeed, 
if  the  season  changed  exactly  with  the  sun's  declination,  all 
effects  of  annual  change  of  temperature  would  be  completely 
eliminated  from  rfi",  as  would  also  in  any  case  any  constant 
error  which  is  a  function  simply  of  the  Sun's  Declination.  It 
is  tlierefore  to  be  expected  that  the  actual  probable  error  of 
this  (luantity  will  conform  more  nearly  to  that  determined  from 
the  residuals  than  in  the  case  of  the  other. 

For  these  reasons  the  value  of  61"  does  not  give  rise  to 
much  di8(!ussion.  The  general  result  from  all  the  observa- 
tories is,  for  61",  when  developed  in  the  form  .v  +  y  T. 

J-  =  -f  0".05 

,/  =  -  0".J>7. 

Obliquity  of  the  ecliptic. 

18.  The  determination  of  the  obliquity  rests  upon  an  essen- 
tially difterent  basis  from  that  of  the  absolute  longitude,  in 
that  it  depends  upon  actual  differences  of  measured  Declina- 
tions, which  differences  are  still  further  complicated  by  the 
fact  that  they  are  necessarily  made  at  opposite  seasons.  A 
more  detailed  discussion  of  them  is  therefore  necessary,  and 
some  modification  may  have  to  be  made  in  the  separate  results 
as  adoi)ted.  The  following  special  circumstances  affecting  the 
observations  are  to  be  taken  into  consideration : 

The  IJRADLEY  Greenwich  results  for  17i)3-'Cli,  are  derived 
from  a  manuscript  communicated  by  Dr.  Auwers,  containing 
the  results  of  his  very  careful  reduction  of  Bradley's  ob- 
served Declinations  of  the  Sun,  which  were  compared  with 
Hansen's  tables.  The  corrections  were  reduced  to  those  of 
Leverrier's  tables  by  being  computed  at  intervals  suffi 
ciently  short  to  permit  of  the  reduction  being  interpolated  with 
all  necessary  precision.  No  reduction  was  applied  either  on 
account  of  the  constant  error  of  the  Declinations  determined 
by  Dr.  Aiwers  himself,  nor  for  reduction  to  the  Boss  system 
of  standard  Declinations,  llence  arises  the  large  value  of  J6 
given  by  these  Declinations.  Consequently  the  value  of  6e  is 
5690  N  ALM 3 


!U 


] 

i         ' 
i 

11               ^ 

pi 

34 


OBSERVATIONS  OF  THE  SUN. 


fl8 


H 


tbat  giveu  immediately  by  the  instrument,  on  tlie  system  of 
reduction  adoj^ted  by  Dr.  Aitwers,  in  which  1  have  supposed 
that  the  I'ulkowa  refractions  were  used. 

From  17(55  to  181G  the  Greenwich  observations  were  made 
with  the  imperfect  qnadrant,  the  Declinations  of  which  are 
subjected  to  an  error  which  is  not  constant.  The  neces- 
sary correctit)ns  are  derived  by  Safford  in  Vol.  ii  of  the 
Astronomical  Papim.  The  corrections  are  those  necessary  to 
reduce  to  Boss's  system,  and  they  vary  with  the  Declination. 
Dence  the  arc  on  which  the  obliquity  depends  is  not  that 
measured  with  the  instrument  itself,  but  that  so  corrected  as 
to  reproduce  as  nearly  as  may  be  the  standard  Declinations. 

From  1812  onward  the  two  mural  circles  were  used.  Up  to 
1830  no  correction  except  the  constant  one  derived  by  Saf- 
ford was  applied  to  the  J^eclinations  as  measured  with  these 
instruments.  Hence  the  arc  of  obliquity  is  that  measured 
with  the  instrument  itself  without  being  corrected  by  the 
standard  stars. 

After  1830  the  Declinations  were  corrected  by  the  tables  for 
Greenwich  given  in  Boss's  paper.  These  corrections  varj' 
somewhat  Avith  the  Declination,  and  they  are  different  also 
for  different  periods.  Hence  we  have  here  a  period  during 
which  the  instrunu'ntal  differences  of  Declination  were  cor- 
rected to  reduce  them  to  the  standard  star- system. 

If  the  standard  system  were  subject  to  no  further  error  than 
a  constant  one,  common  to  all  J)ec]inations  within  the  zodiac, 
which  common  correction  would  be  subject  to  a  uniform  change 
with  the  time,  this  system  would  doubtless  be  the  best  one  to 
adopt  in  or<ler  to  obtain  the  secular  variation  in  the  obliquity 
of  the  ecliptic.  But,  as  a  matter  of  fact,  the  standard  Decli- 
nations are  simply  the  mean  results  of  Declinations  measured 
with  different  instruments.  It  is,  therefore,  a  <iuestion  whether 
we  shall  get  any  better  results  by  applying  reductions  to  a 
standard  system  than  we  should  get  by  simply  taking  the 
mean  of  the  instrumental  results,  because  the  system  is  itself 
only  a  mean  of  such  results.  It  is  true  that  the  standard  sys- 
tem depends  on  more  instruments  than  the  obliquity,  though 
not  on  better  ones;  but  it  is  also  to  be  considered  that  the 
JL'eductions  in  the  case  of  the  Sun  may  be  different  from  those 


18,  19] 


OBLK-iUlTY  OF   ECLIPTIC. 


35 


in  the  case  of  the  stars,  owiiijj  to  the  Very  ditiereut  eoiulitions 
in  which  the  observations  are  made. 

Another  troublesome  \mut  arises  fron)  the  refraction  used 
in  the  reductions.  The  effect  of  refraction  is  always  to  make 
the  measure«l  obliquity  less  than  the  actual  one;  the  correc- 
tion to  tlie  obliquity  ou  account  of  refraction  is  therefore  a 
positive  (|uantity,  which  is  a  minimum  f(U'  an  observatory  at 
the  equator  and  increase  e<jually  towards  each  pole.  Some 
values  of  the  obliquity  were  derived  from  Bessel's  refractions 
of  the  Tabula;  Jieffiomontame,  and  others  from  the  Tulkowa 
tables.  Since  the  secular  variation  of  the  obli(|uity  is  more 
important  than  the  absolute  value  of  the  (juantity,  it  is  essen- 
tial that  the  standard  to  which  all  determinations  of  the  ob- 
liquity are  reduced  should  be  as  nearly  as  possible  the  sauu', 
and  therefore  that  the  same  refraction  should  be  used.  But  in 
reductions  to  stand.ard  star  places  we  meet  with  tlie  addi- 
tional complication  that  the  differences  in  the  constant  of 
refraction  might  be  wholly  or  partially  eliminated  by  the 
reductions  to  a  standard  system.  It  would  therefore  be  a  dif- 
ficult ipiestion  how  far  we  should  modify  the  values  of  St  on 
account  of  the  use  of  different  tables  of  refraction. 

To  avoid  all  these  difficulties  I  have  Judjied  it  best  to  make 
the  obliquity  depend  mainly  upon  absolute  measures,  the 
reductions  being  made  with  the  Pulkowa  refractions. 

Effect  of  refraction  on  the  obliquity. 

19.  The  determination  of  the  average  or  most  probable  effect 
ou  the  obli«iuity  produced  by  using  the  Pulkowa  refractions, 
instead  of  those  of  the  Tahuhv  Kegiomontana',  is  easily  deter- 
mined. We  divide  the  ecliptic  into  a  number  of  cjjual  arcs 
throughout  the  year,  and  by  equations  of  condition  express 
differences  of  refraction  in  terms  of  differences  of  Declination, 
and  hence  differences  of  obli«piity.  We  thus  find  that  at 
certain  latitudes  where  observations  were  made,  and  where 
Bessel's  refractions  were  used  in  the  reduction,  the  follow- 
ing corrections  are  necessary  to  reduce  the  oblifjuity  to  the 
ones  given  by  the  Pulkowa  refractions: 

Pulkowa;         c/j  =  o9o.8;  ^e  —  -  0".325 
Greenwich ;      ^  =  .jfo.o;  Je  =  -  0".20 
Washington;  <p  =  3So.9;  Jf  =  -  0".125 


li 


!    ? 


I 


'M  OBSERVATIONS   OF  THE   SIN.  [19 

Hence  I  conclude  that  for 

Dor  pat;  Jf  =  -  0".29 
Kiinijjsber^;  Jf  =  —  0".!i<» 
Cambridge;  J*  =  -  0".21 
Cape  Town ;  Jf  =  -  0".1L' 

The  corrections  to  tlie  obli(iiiity  thus  derived,  depending 
mainly  on  direct  instrumental  measurement,  and  reduced  to  the 
Pulkowa  refractions,  are  desipiated  as  (Vf.  The  results  for  this 
quantity  are  friven  in  tlie  last  column  of  the  several  tables. 

In  the  case  of  Bradley's  Greenwich  results,  I  have  taken 
as  (Ve  Dr.  Aitwers'.s  results  unchanged,  assuming  in  the 
absence  of  any  specific  statement  that  he  has  used  the  I'ul- 
towa  refraction  tables. 

In  the  case  of  ^Faskylene's  observations,  1  have,  by  excep- 
tion, used  them  as  reduced  to  the  standard  stfir-system, 
because  we  have  no  other  results  at  these  times,  and  the  en  or 
of  his  instrument  is  so  stror.gly  shown  that  it  would  not  do  to 
use  the  results  unchanged.  It  will  be  seen,  however,  tliat 
small  weights  are  assigned,  and  that  the  weights  diminish 
towards  the  end  of  the*series. 

lu  the  case  of  the  Greenwich  observations  from  1812  to 
about  1834,  no  change  has  to  be  made,  as  the  results  are  gen- 
erally or  always  purely  instrumental,  and  Pulkowa  refractions 
are  used  in  Safford's  work. 

From  1835  onward  I  have  de])eiuled  mainly  on  certain  cor- 
rected Greenwich  reductions.  First,  for  (Vf,  I  have  used  the 
results  given  by  Mr.  (.'hristie  in  his  very  valuable  paper  on 
the  Greenwich  Declinations,  in  M.  R.  A.  S.,  Vol.  xlv,  where 
the  Declinations  from  1830  to  1879  are  reduced  on  a  uniform 
system.  Later,  1  have  adopted  the  corrected  results  given  in 
Appendix  III  to  the  Greenwich  observations  for  1887.  In 
each  case  the  result  has  been  reduced  to  the  Pulkowa  refrac- 
tions. 

The  Paris  results  rest  on  a  different  basis  fiom  the  others, 
in  that  the  zero  point  of  the  instrument  depends  wholly  upon 
liEVi-^iRRiER's  Declinations  of  the  stars,  and  I  fear  it  was  not 
always  axjcurately  determined.  Observations  near  the  winter 
solstice  are  mostly  referred  to  one  set  of  stars;  those  near  the 


19J 


OBIyiyUlTY   OF  ECLIPTIC. 


37 


suniiiHT  to  another  set,  the  error  of  which  may  be  systemat 
ically  (litt'eient.  (Certain  it  is  that  the  results  during  the  early 
years  were  very  diseordant.  The  weifjfhts  as  {;iveu  in  the  table 
are  those  assigned  a  priori,  without  sutlieieut  reference  to  the 
discordance  of  the  older  results.  I  have  felt  constrained  to 
evade  a  decision  as  to  their  treatuuMit  by  entirely  omitting 
their  results  in  the  Hnal  discussion. 

In  the  case  of  sonie  other  observatories  it  was  difficult  to 
determine  exactly  what  refractions  had  been  used  in  each 
S|)ecial  case  and  what  reductions  should  be  made.  1  have,  how- 
ever, determined  the  corrections  in  the  best  way  1  was  able. 

A  i)re(!ise  determination  of  the  secular  change  in  the  ob- 
liquity is  of  ujore  importance  for  our  present  object  than  a 
precise  determination  of  its  amount.  Hence  a  series  of  obser- 
vations extending  through  a  long  jjeriod  of  time,  and  umde  on 
a  uniform  system,  has  an  advantage  over  a  uumber  of  isolated 
values,  in  that  any  constant  error  with  which  it  may  be 
affected  will  be  eliminated  from  the  secular  variation.  Possi- 
ble constant  differences  between  the  determinations  of  the 
various  observatories  at  different  epochs  will  vitiate  the  sec- 
ular variation,  but  the  probable  amount  of  this  error  may  be 
diminished  by  using  a  number  of  separate  determinations, 
such  as  are  i)resented  in  the  preceding  table.  In  the  Greftn- 
wich  transit  circle  we  have  a  very  uniform  series,  extending 
over  a  period  of  forty  years,  but  giving  results  systematically 
different  from  other  determinations.  This  series  gives  t'ov  the 
correction  to  the  obliquity: 

Transit  Circle,  18 17-'91 : 


6'€  =  -  0".ll  i  0".0C  +  (0".21  ±  0'  .40)  T 


(a) 


Here,  in  view  of  the  uniformity  of  method  and  leduction, 
we  may  regard  the  mean  error  of  the  centennial  variation  from 
the  discordance  alone  as  a  fair  approximation  to  the  probable 
mean  error.  It  will  be  seen  tliat  I  have  here  induced  four 
years  (1847-'r)0)  of  the  Mural  Circle  results. 

Continuing  the  Greenwich  series  backward,  the  question 
arises  whether  we  can  regard  the  results  of  the  mural  circle 
from  1812  to  1850  as  comparable  with  those  of  the  transit  circle. 


38 


OBSERVATIONS  OF  THE  SUN. 


[19 


There  is  certainly  nothing  in  the  table  to  indicate  any  system- 
atic ditt'erence.    From  the  combination  of  the  two  we  have^ 

M.  C.audT.  C,  1812-T)0: 

d'e  =  -  0".08  ±  0".(>r)  +  (4-  0".14  ±  0".23)  T  (1850)  .  .    (b) 

Here  the  mean  error  is  naturally  smaller  than  in  the  case  of 
tie  transit  circle  alone,  but  is  now  more  subject  to  possible 
systematic  difference  between  the  two  instruments. 

If  we  now  go  back  to  Bradley,  we  meet  with  the  very  diffi- 
cult question,  whether  we  should  regard  his  results  as  best 
comparable  with  the  modern  Greenwich  observations,  or  with 
modern  observations  in  general.  If  we  assume  that  the  differ- 
ence between  the  Greenwich  and  other  modern  results  is  due 
to  any  cause  which  has  remained  unchanged  since  Bradley, 
we  should  reach  one  conclusion;  otherwise,  we  should  reach 
the  othci'.  The  result  of  combining  all  Greenwich  observa- 
tions, with  the  weights  as  assigned,  is — 

(Ve  =  -  O'Ml  +  0".r)0  T (0) 

In  this  combination  I  have  used  the  weak  results  of  JNIaske- 
LYNE,  with  the  small  weights  assigned,  although  they  d  4>end 
wholly  upon  the  standard  declinations  of  stars.  In  view  of 
the  discordance  between  Bradley's  two  results,  tliis  seems 
the  only  admissible  course. 

Next  in  the  length  of  time  which  they  include  come  the  Paris 
observations,  of  which  the  results,  with  the. weights  assigned, 
are — 

(Jf  =+0".01-0".3GT 

I  give  this  result  in  order  that  nothing  may  be  omitted. 
Undue  weight  has  probably  been  assigned  to  the  earlier 
determinations;  in  any  case  the  method  of  deriving  it  from 
the  original  observations  is  so  objectionable  that  no  further 
use  is  made  of  it.  A  satisfactory  discussion  of  the  observa- 
tions would  require  a  complete  redetermiuation  of  the  zero 
points  of  the  instrument  from  fundamental  stars. 


19,  20 1         DISCUSSION  OF  RESULTS   OF  OBLIQUITY.  39 

If  we  omit  the  (ireenwich,  Paris,  and  Palermo  results,  and 
combine  all  the  others  into  a  sinj^le  set  of  eqnations  of  condi- 
tion, we  have  the  eijuationa  and  resnlts: 

3fi.<)j+0.2«J»/=  -  14".37 
U.LM)  +  1.S8    =  +    1"-01 

y=+  0".r)9 

Here  .»•  is  the  valne  of  rf'f  for  18<»0,  and  y  its  centenniiil  varia- 
tion.   Transferring;  the  epoch  to  .1850,  as  nsual,  the  result  is — 


6'f=  -  o".4.")  +  (>"..v.rr 


e?) 


No  reliable  mean  error  can  be  computed,  owing  to  systematic 
errors.  In  view  of  these,  one  mode  of  treatment  would  be  to 
form  eijuations  of  coiuTition  in  whic  h  a  possible  systematic 
error  at  each  observatory  wouhl  appear  as  one  of  the  unknown 
quantities.  By  this  process  we  should  j^et  the  same  result 
for  the  secular  variation  as  if  we  made  an  independent  determi- 
nation from  the  work  of  eaciii  ol)8ervatory.  At  most  of  the 
observatories  the  period  throuj^h  which  the  observations  are 
made,  with  one  instrument  an<l  on  an  unchanged  plan,  is  too 
short  to  render  such  a  course  advisable. 

As  a  last  combination,  we  shall  combine  the  earlier  (Ireen- 
wich  results,  up  to  1810,  with  Palermo  and  with  all  the  modern 
results  except  Paris,  first  dividing  the  weights  of  the  Green- 
wich results  by  2.     We  then  have  the  equations — 

39.8  .r-  1.82  J/ =  -  17".  12 
-  1.8     -I-  3.47     =  +    2".t>9 


X  =  -  ()".40 
y=+  0".(M 


(«) 


Concluded  results  for  the  oblir'ty. 

20.  The  data  on  which  these  various  results  for  the  obliquity 
rest  show  the  following  noteworthy  features : 

(1)  That  the  correction  given  by  the  modern  Greenwich 
instruments,  mural  an«l  transit  circles,  ia  markedly  greater 


40 


OBSERVATIONS   OF  THE   HUN. 


(20 


h 


tban  that  given  by  otlna-  iiioderii  obMervatioiis.  This  may  be 
most  ])hiUHibly  attributed  to  the  atinusplieric  coiKlitioiis 
within  the  observing  room. 

(2)  The  niinnteness  of  tlie  change  of  tlje  correction  given 
by  these  instrnnionts  during  nearly  eighty  years.  To  this 
cinnnnstance  is  due  the  snuiUness  of  the  centenuial  variation, 
0".itO,  found  from  tlie  totality  of  the  (heenwich  observations. 
A  comparison  of  Bkadley  with  the  mean  of  the  T.  C.  results 
only  would  have  given  a  change  of  0'M)7  in  117  years,  or  a 
centennial  change  of  about  0".8(). 

The  long  periotl,  uniformity  of  plan,  and  systematic  devia- 
tion of  the  modern  (irecuwich  observations  lead  me  to  consider 
them  as  forming  a  series  distinct  from  all  others.  We  have 
therefore  the  following  two  completely  independent  determi- 
nations of  the  centennial  variation: 


(1)  Modern  Greenwich  results:  y  = -\-  0".14  =k  0".23 

(2)  All  other  results  +  0".(i5 

To  the  latter  no  reliable  mean  error  can  be  assigned.  To 
jutlge  its  reliability  we  may  compare  it  with  the  results  {a),  (c), 
and  (d) — 

Greenwich  T.  C,  alone,  +  0".21  ±  ()".46 

Greenwich  obseivations  in  general,  4-  0".50 
Miscellaneous  modern  observations,  +  0".51) 

We  may,  it  would  seem,  fairly  give  double  weight  to  the 
result  (2),  thus  obtaining,  as  the  detiuite  result  from  observa- 
tions of  the  Sun  alone: 

Correction  to  Leverrier's  centenuial  variation  of  the  obliq- 
uity of  the  ecliptic  ( —  ■17".594) 

■f  0".48  ±  0".30 

the  mean  error  being  an  estimate  from  the  general  discordance 
of  the  data. 
For  the  coustaut  jiart  of  the  correction  I  take — 


(y*(1850)  =  -0".30 


21 


Sl'MMAEY  OP   KKSULTS. 


41 


tSumiiiartf  and  vAtmpariHon  of  rcHHltn. 

21.  From  what  procodcs  wo  liiivo  tlio  lollowiii{i  iim  tin*  values 
of  the  unknown  quantitien,  and  of  tiieir  socular  variations,  as 
given  by  ob-servatiouH  of  the  Sun  alone. 


Value  for 
1S50. 

fir"  =  H-  O'MO    I    (V'.OM 

c>'{S7T"-\.a)  =       0".0()  4-.  0".(>7 
8\"-\.a  =  -  0".(>2 

61"  =  -f  0".(»5  4r  0".12 
rff  =  -  0".aO  4:  0".15 
<r  =  -  ()'M)7 


t.'cnl. 
vnr. 

+  0".23  4:  0".10 
■f  ()".33  I   0".12 

-  {)"M 

-  (>'M)7  1:  0".23 
4-  0".48  -Jtz  0".30 
+  ()".34 


No  estininte  of  the  probable  errors  of  these  (luantities  would 
be  useful  which  did  not  take  account  of  the  Hysteniatic  dif- 
ferences between  the  results  of  ditterent  observatories.  We 
have  therefore  formed  the  mean  outstanding  residual  correc- 
tions given  by  the  several  observatories,  us  shown  in  the 
tables  which  foH(»w.  Originally  the  scale  of  weights  used  for 
the  Greenwich  observations  did  not  correspond  to  that  for  the 
other  observatories;  they  were,  therefore,  divided  by  2.  As 
used  below,  however,  the  change  has  been  made  in  the  case 
of  61"  by  multiplying  all  the  weights  of  the  other  observatories 
by  2,  and,  in  the  case  of  6s,  by  dividing  the  Greenwich  weights 
by  2. 

The  correction  to  the  obliquity  depends  solely  on  6'e;  but 
the  comparison  has  also  been  made  with  the  values  of  df, 
which,  it  will  be  remarked,  differ  from  the  others  in  that 
account  is  taken  of  the  supposed  variation  of  the  systematic 
correction  with  the  declination.  Jt  is  noteworthy  that  the 
results  are  somewhat  more  accordant  when  this  correction  is 
omitted  and  jiurely  instrumental  errors  are  used  for  the 
obli(iuity. 

The  mean  errors  giveu  in  the  preceding  summary  of  results 
are  derived  from  the  discordances  in  question,  and  may  be 
regarded  as  substantially  real. 

No  use  was  made  of  the  Paris  results  for  61"  and  6e  for 
the  reason  that  they  depend  on  declinations  referred  to  star 


11= 


42 


OBSFiiVATlONS  OF  THE  SUN. 


[21 


places  wbi<'li  may  be  att'ected  by  differences  in  different  Right 
Ascensions.  They  are,  liowever,  retained  in  the  table  to  show 
the  ivinounts  of  outstanding  discordance. 


Outstandirnj  mean 

residual  corrections  to  qnantities  depending 

on 

the  tSun 

'«  Bight  A 

scension. 

8e" 

e'Sn" 

2w 

Greenwich 

+  0".09 

-  0".03 

o4.5 

Paris 

-  0".09 

+  o".n 

17 

(.'anibridge 

+  0".02 

0".00 

16 

\\'asliington 

-  0".05 

-0".1L' 

24 

Kbnigsberg 

-  0".08 

+  0".()8 

12 

Oxford 

+  0".0<i 

-f  0".02 

8 

Pulkowa 

-  0".15 

+  0".22 

e 

Dorpat 

-  0".10 

-  0".03 

4 

Cape 

-   0".16 

-O'Ml 

4 

Strassburg 

+  0".05 

-  0".03 

3 

^ITean  errors 

for 

weight  unity    ^i  = 

zt  0".34 

±  ()".39 

!Mean  error  of  .r 

±  0".03 

A,  0".03 

^Ican  error  ofy 

±  O'MO 

±  0".12 

Ovtstandlng  mean  residual  corrections  to  qnaiitities  depending 
on  the  8un''8  Declination. 


dV 

w 

•     Se 

«' 

S'£ 

Greenwich 

-  0"M 

(14 

+  0".31 

29.0 

-f  0".17 

Paris 

+  0".45 

0 

+  0".31 

0' 

Palermo 

-  0".30 

0 

-  0".20 

0.8 

-  0".20 

Cambridge 

-  0".05 

8 

4-  0".35 

4 

+  0".14 

Washington 

+  0".07 

24 

-  0".22 

12 

-  0".29 

Kiinigsberg 

-  0".20 

10 

+  0".31 

5.5 

0".00 

Oxford 

■f  0".14 

14 

4-0".19 

1.4 

-  0".01 

Pulkowa 

+  0".12 

8 

-  0".13 

4 

-  0".13 

Dorpat 

+  0".75 

G 

-  0".49 

3 

-  0".04 

Capo 

-  o".3r. 

8 

+  0".10 

4 

-  0".02 

Leiden 

+  0".10 

8 

+  0".1V 

o 

-  0".06 

Strassburg 

-  0".26 

4 

+  0".08 

4 

+  0".25 

f  for  wjight  unity 

±  0".81 

±  0".74 

±  0".00 

'.«*•.., 


CHAl'TEK  HI. 

RESULTS  OF   OBSERVATIONS  OF    MERCURY,  V.    v  JS,  AND 

MARS. 

Elements  lu.opied  for  correction. 

22.  We  first  give  an  outline  of  the  method  of  expressing  the 
observed  corrections  to  the  Right  Ascensions  and  Declinations 
of  each  of  the  planets  as  linear  functions  of  the  correlations  to 
the  tubular  elements.  This  linear  function  forms  the  first 
member  of  the  equation  of  condition  in  its  original  form,  and 
the  observed  correction  forms  its  second  member. 

Let  us  put — 
It,  >•,  the  radii  vectores  of  the  Earth  and  planet; 
L,  the  Sun's  true  longitude; 

J,  the  inclination  of  the  orbit  of  the  planet  to  a  plane 
passing  through  the  Sun's  center  parallel  to  the 
plane  of  the  Earth's  equator; 
N,  the  Right  Asceusion  of  the  ascending  notle  of  the 
orbit  on  this  plane; 
.     U,  the  argument  of  helioc  entric  declination  of  the  planet 
or  its  angular  heliocentric  distance  from  the  node 
on  the  etjuator; 
rf,  (J,  the  geocentric  Right  Ascension  atul  Decliimtiou  of 
the  planet, 
e,  the  obliquity  of  the  ecliptic; 

We  shall  then  have — 

.r  =/(>'.  R.  L.  J.  X.  U.,  K) (a) 

lAu-  the  correction  to  the  tabnlar  Right  Ascension  arising 
from  symbolic  corrections  to  t!'  >se  seven  quantities,  we  have 
the  e(iuation — 


m 


43 


(  . 


44 


MER(!URY,   VENUS,   AND  MABS. 


[22 


I 


with  a  similar  e(|uation  for  the  declinatiou,  formed  from  this  by 
writing  6  for  a. 

The  relations  by  which  these  two  equations  are  derived,  as 
well  as  the  expr«»s8iou8  for  the  difterential  coefticieuts  they 
contain,  are  given  very  fully  in  A.  P.,  Vol.  II,  Part  I,  to  which 
reference  may  be  made.  The  corrections  dN  and  6U  are  not, 
however,  the  most  convenient  ones  to  choose.  It  will  be  found 
in  the  paper  faiuded  to  that  they  have  been  transformed  by 
measuring  the  longitude  in  orbit  of  the  planet  and  that  of  the 
perihelion  from  an  arbitrary  point  in  the  orbit.  As  to  this  very 
convenient  device  in  celestial  mechanics,  it  is  to  be  remarked 
that  the  "departure  point"  always  disappears  from  the  final 
e(|u<itions  which  determine  the  position  of  the  i)lanet.  We 
may,  in  fact,  make  abstraction  of  it  by  considering  that  its 
introduction  is  equivalent  to  the  following  simi)le  linear  trans- 
formations. 

We  put 

w,  the  distance  from  the  node  to  the  perihelion; 
/,  the  true  anomaly; 
g,  the  mean  anomaly. 
7T,  the  longitude  of  the  perihelion ; 
I,  the  mean  longitude  of  the  planet; 
V,  its  true  longitude; 

these  longitudes  l)eing  counted  from  the  departure  point. 
Then  we  have  the  relations — 


I 


Hence, 


SU  .=  6\v  -I-  6/  .=  dr  -  cos  JdN 
d\\  =  67T    —  cos  J(5N 
61  =  Stt    +  6g 

d7r  =  6TJ-\-  cos  JdN  -  6/ 


(2) 


(3) 


The  elements  finally  adopted  for  correction  by  the  equations 
of  condition  were — 

I.  n.  e.  J.  N.. 


J 


22,  231 


ELEMENTS  ADOPTED  FOR  CORKECTION. 


45 


The  value  of  a,  the  mean  distance,  »  known  Avitli  such  pre- 
cision that  its  correction  need  not  enter  into  the  equations  of 
condition.    The  latter  are  formed  by  substituting  in  (1) 


dV  =(l  -  ^Y\  Srr  +  -Ide  +  ^Y-dl  -  cos  JfJN. 
V  d(iJ  de  (hj 


o,        dr  J.    ,   dr  ^,      dr  ^ 
de  d(j  dy 


(4) 


The  coefficients  of  each  equation  of  condition  from  the  Kight 
Ascension  thus  become — 


Coefficient 

of  63 

(( 

"    tfX 

<( 

"       6£ 

« 

^'      61 

« 

'•    6n 

(( 

'<     6e 

da 

W 

da 

da 

rfN 

— 

CO!! 

1  J 

du 

da 

de 

da 
d\5 

do 

+ 

da 
dr 

dr 
dy 

(5) 


da  /  ^  _  df\  _  da  dr 
dU\         dnj       dr  dg 
da  df      da  dr 
(W  de  "^  dr  de 


In  the  second  members  of  the  equations  a  is  regarded  as 
a  function  of  the  seven  quantities  {a\.  as  is  also  6,  for  which 
a  similar  equation  is  to  be  formed. 

The  correj'tions  of  the  solar  (M'rentricity,  peiihelion,  and 
mean  longitude  w  3re  also  intri        cd  1)y  putting  in  (1) 


(«) 


Introduclinn  of  the  masses  of  Venus  and  Mer<  xry. 

2.'i.  The  correction  to  the  mass  of  Venus  was  introduced 
by  taking  the  tabular  perturbation  produced  by  Venn-  on 
the  geocentric  i)lace  of  the  planet  at  the  mean  dn !  each 

equation  as  the  coefficient  of  the  unknown  quantity  to  be 
determined.     In  computing  these  perturbations  regard  was 


46 


MERCURV,  VENUS,  AND  MAE8, 


[23,  24 


had  to  the  action  of  Venus  on  the  Earth  as  well  as  on  the 
planet.  On  this  system  the  unknown  quantity  finally  found 
would  be  the  factor  by  which  the  adopted  mass  of  the  planet 
must  be  multiplied  in  order  to  give  the  correction  of  that  mass. 

It  has  already  been  remarked  that  the  mass  of  a  idanet  can 
not  be  deteruuncd  free  from  systematio  error  by  observations 
made  upon  the  planet  itself.  Hence,  the  mass  of  Venus  can 
be  determined  oidy  from  observations  of  Mercury  and  Mars, 
and  that  of  Mercury  only  from  observations  of  Venus  and 
Mars.  But  the  mass  of  Mercury  is  so  minute  that  it  would  be 
useless  to  attempt  to  deternjine  it  from  observations  either  of 
the  Sun  or  Mars.  It  was  therefore  determined  solely  from  the 
periodic  perturbations  of  Venus. 

It  has  hai)i>ened  that  the  mass  of  Venus  could  not  be  deter- 
mined in  a  reliable  way  from  observations  of  Mars,  owing  to 
a  defect  in  the  theory  of  the  latter  planet,  which  I  shall  men- 
tion hereafter,  ai  1  have  not  yet  had  time  to  correct.  Practi- 
cally, therefore,  the  mass  of  Venus  is  determined  only  from 
observations  of  the  Sun  and  of  Mercury,  and  that  of  Mercury 
from  observations  of  Venus. 

Correction  of  equinox  and  equator. 

24.  Could  all  the  observations  be  directly  referred  to  a 
visible  eijuinox  and  e<|uator,  the  corrections  above  enumerated 
Mould  have  been  the  only  ones  which  it  was  necessary  to 
include  in  the  equaticis  of  condition.  But,  as  a  nmtter  of 
fact,  the  observations  were  all  referred  to  an  assumed  system 
of  Hight  Ascensions  and  Declinations  of  standard  stars — my 
own  system  in  Hight  Ascension  and  Boss's  in  Declination. 
We  must  therefore  introduce  two  additional  unknowns  into 
the  equations,  which  1  have  repn^sented  in  the  following  way: 

a,  the  common  error  of  tlie  adopted  Right  Ascensions. 
6,  the  common  error  of  Boss's  1  )eclinations. 

The  first  quantity  will  appear  only  in  the  equations  derived 
from  observed  Right  Ascensions  and  the  second  only  in  the 
equations  derived  from  Declinations,  the  coeflScient  being  unity 
in  each  case. 


24] 


COREECTION  OF  EQUINOX  AND  EQUATOR. 


47 


That  the  value  of  6  found  iu  this  way  should  be  regarded 
as  a  correction  to  the  Declinations  of  the  equatorial  stars  will 
appear  by  the  following  considerations.  The  mean  heliocen- 
tric orbit  of  a  planet  as  projected  on  the  celestial  sphere  is 
undoubtedly  a  great  circle.  On  the  other  hand,  in  view  of  the 
systematic  discordance  always  found  to  exist  in  measures  of 
absolute  Declinations  near  the  equator,  and  of  tlu^  fact  that 
these  absolute  Declinations  depend  upon  assumed  constants 
and  laws  of  refraction,  which  are  necessarily  aft'ected  witli 
greater  or  less  uncertainty,  and  are  otherwise  subject  to 
systematic  errors,  instrumental  or  personal,  of  an  obscure 
character,  but  strongly  shown  by  a  comparison  ot  tlie  Declina- 
tions deiived  from  the  work  of  different  observatories,  it  can 
not  be  assumed  that  these  Declinations  are  free  from  sys- 
tematic error.  Now,  in  one  circle  ot  Decimation,  say  the 
e(iuator,  we  may  expect  that  the  error  will  be  nearly  constant 
around  the  sphere,  since  the  causes  of  error  will  generally  be 
nearly  constant  for  any  one  Declination.  This  conclusion  is 
confirmed  by  a  comparison  of  the  best  star  catalogues. 
Moreover,  between  the  zodiacal  limits,  the  error  in  each  par- 
ticular case  is  not  likely  to  diftei  very  greatly  from  the  error 
at  the  equator.  Even  if  the  difference  should  be  considerable 
the  various  values  of  the  error  of  the  different  Declinations 
must  have  a  certain  mean  value,  so  that  in  the  case  of  each 
particular  star,  or  each  region  of  the  lieavens,  we  may  conceive 
the  actual  error  to  be  divided  into  two  parts — one  the  mean 
value  in  (juestiou,  and  the  other  the  deviation  from  this  mean. 
The  latter  is  probably  smaller  chixn  the  former,  and  in  any 
case  can  not  very  well  be  determined  from  observations  of  the 
l)lanets.  But  the  condition  that  the  planet  moves  on  a  great 
circle  of  the  sphere  admits  of  the  mean  value  being  deter- 
mined with  great  precision.  It  should,  therefore,  be  included 
in  this  equations  of  condition. 

The  value  of  <y,  the  common  error  of  all  the  Kight  Ascen- 
sions, can  obviously  not  be  determined  from  the  equations  in 
Right  Ascension  alone,  because  the  only  result  that  such 
observations  can  give  us  would  be  the  values  of  the  Right 
Ascensions  referred  to  some  assumed  equinox.  The  coefficient 
of  a  would  therefore  completely  disappear  from  the  equations 


48 


MKKCURY,   VKNUS,   AND  MARS. 


[24 


of  condition  in  Right  Ascension.  But  since  the  same  unknown 
(juantities  are  introduced  into  tlie  e<|uations  of  condition  in 
liight  Ascension  and  in  Declination,  the  re<iuirement  that  the 
two  sets  of  e(|uati(ms  shall  give  coinnion  values  of  these 
(luantities  does  away  with  this  indetermination  and  enables 
determinate  values  to  be  found.  In  fact,  this  method  does  not 
dift'er  in  principle  from  that  usually  adopted  in  deriving  the 
Right  Ascensions  of  stars  from  observations  of  the  Sun.  The 
latter  consists  in  deriving  the  Sun's  absolute  longitude  from 
observations  of  its  Declination  and  }ibso)ute  Right  Ascensions 
of  the  stars  by  comparing  them  with  the  Sun.  In  the  same 
way  we  may  consider  that,  in  observations  of  the  planet,  the 
Sun's  absolute  longitude  is  derived  from  observatiiuis  of  Decli- 
nations of  the  planet,  and  then  a  comes  out  from  the  observa- 
tions in  Right  Ascension. 

I  have  deemed  it  absolutely  necessary  that  all  the  equations 
of  condition  should  be  solved  by  the  method  of  least  squares. 
liy  this  method  alone  can  the  results  of  the  observations  as 
regards  separate  values  of  the  elements  and  constants  be  prop- 
erly brought  out.  But  the  work  of  constructing  and  solving 
a  system  of  nine  thousand  equations  of  condition,  each  involv- 
ing twenty  unknown  quantities,  would  be  extremely  laborious, 
and  might  even  require  a  century  for  its  completion,  if  done  in 
the  usual  way.  It  was  therefore  necessary  to  adopt  every 
device  by  which  the  labor  could  be  reduced  to  a  minimum. 
One  device  was  the  dropping  of  all  superfluous  decimals  in  the 
coetticients  of  the  equations.  Since  the  errors  thus  produced 
would  be  purely  accidental,  it  follows  that  if  the  sum  of  tie 
produ<'ts  obtained  by  multiplying  the  value  of  each  unknown 
quantity  by  the  error  of  its  coeiticient  in  the  eijuation  of  con- 
dition is  but  a  small  fraction  of  the  necessary  probable  error 
of  the  absolute  term,  no  serious  harm  will  result  from  the 
errors  of  the  coetticients. 

Another  device  was  the  construction  of  tables  for  finding 
the  coetticients.  Such  tables  relating  to  Mercury  and  Venus 
are  found  in  Vol.  II,  Part  I,  of  the  Astronomical  Papers. 
These  tables  are,  however,  only  given  for  one  mean  anomaly  in 
each  case,  and  therefore  require  comi)utation8  dependent  on 
the  value  of  the  other  anomaly.    They  were  therefore  extended 


f24 


24,  25]       INTRODUCTION  OP  SECULAR  VARIATIONS.  49 

to  tables  of  double  entry,  so  that  the  value  of  the  derivatives 
of  the  ge<K'entric  Kight  Ascension  or  Declination  at  any  epoch 
could  be  taken  from  the  tables  at  sight.  The  arguments  were 
the  mean  anomaly  of  the  planet  and  the  day  of  the  year  at 
which  the  planet  last  passed  through  its  perihelion. 

Introduction  of  the  secular  rar'mtiom. 

25.  When  the  equations  of  condition  are  formed  on  the  plan 
just  set  forth,  the  unknown  quantities  will  be  the  corrections 
to  the  elements  or  to  the  mean  longitude  at  the  date  of  each 
eqnatiem.  But  every  one  of  the  unknown  cjuantities  whiclr 
have  been  enumerated,  the  correction  of  the  masses  excepted, 
is  subject  to  a  secular  variation.  Hence,  instead  of  the 
unknown  quauiities  heretofore  defined,  we  introduce  two 
others,  the  one  the  value  of  this  unknown  at  some  assumed 
mean  ei)Och,  which,  for  reasons  already  set  forth,  must  first 
be  determined  from  the  observations;  the  other  the  secular 
variation  in  a  unit  of  time.  The  unknown  «|uantities  which 
have  been  enumerated  make  twelve  for  each  equation  of  con- 
dition. Kleven  of  these  are  subject  to  a  secular  variation,  so 
that  if  the  secular  variations  were  introduced  into  the  original 
equations  of  condition  they  would  each  have  twenty-three 
unknown  quantities. 

The  following  device  was  employed  to  reduce  to  a  mininmm 
the  work  of  introducing  and  determining  the  secular  variations 
of  the  various  elements : 

Firstly,  t{",e  whole  time  covered  by  the  observations  was 
divided  into  periods,  never  exceeding  ten  years,  except  when 
the  observations  were  very  few  in  number,  or  entitled  to  but 
small  weight.  It  was  then  assumed  that  no  error  would  arise 
from  supposing  the  value  of  the  unknown  quantity  to  be  the 
same  throughout  the  period  as  it  was  at  the  mid-epoch  of  the 
l)eriod.  The  maximum  absolute  ernu"  thus  arising  would  be 
the  secular  variation  during  half  the  length  of  the  period,  and 
the  m  jan  eiTor  the  secular  variati<m  during  one-fourth  of  the 
period;  but  actually  the  effect  of  even  this  error  would  be 
almost  entirely  r.ullified  by  the  combination  of  positive  and 
negative  coeflBcients  throughout  each  period. 
569()  N  ALM 4 


11 


M  MERCURY,  VENUS,  AND  MARS. 

Let  US  now  put 


[25 


^,  y. 


•      •      • 


the  corrections  to  tbe  elements  at  any  epoch,  t. 
Let 


ax-\-hy-\-cz-{-.    .    . 


n 


be  an  equation  of  condition  between  these  quantities  at  this 
epoch.  From  a  system  of  such  equations,  extending  through  a 
period  numbered  i,  during  which  x,  y,  etc.,  may  be  considered 
as  constant,  we  derive  normal  equations  of  the  form — 


[aa],a;-^  \ab\fy-{- 
[ab],  X  +  [hhl  y  4- 


=  [aw], 


(1) 


which  I  shall  call  partial  normal  equations,  and  which  we 
miglit  solve  so  as  to  obtain  the  values  of  a?,  2/,  etc.  This  solu- 
tion is  not,  however,  necessary.  The  values  of  the  unknown 
quantities  being  really  of  the  general  form — 


X  =  Xo-\-  x'  t 

P'^Vo  +  y'  t 


(2) 


we  may  imagine  these  values  substituted  in  the  normal  equa- 
tions (1),  the  value  t,  of  t  for  the  mean  epoch  of  the  period 
being  substituted  for  t. 

Let  us  now  suppose  that  we  introduce  the  quantities  Xo,  yoj  •  •  > 
x',  y',  .  .  into  the  original  equations  of  condition,  using  for  t 
the  value  r„  which  pertains  to  the  mean  epoch  of  the  period. 
Our  equation  of  condition  will  thus  become — 


a^o  +  byo  + 


+  ar^x'  +  bT,y'  + 


n 


(3) 


If  from  a  system  of  conditional  equations  of  this  form  we 
form  the  normal  equations  for  all  the  unknown  quantities,  the 
results  will  be  these : 

Partial  normal  equation  in  Xo', 

[aa],a?o -j-  [ab],  j/o  +  .  .  +  r,  [aa],x'  +  r, [ab],y'  +  .  .  =  [an],  (4) 


(1) 


25]  INTEODUOTION  OF  SECULAR  VARIATIONS.  81 

Partial  normal  equation  iu  x'  j 

T,  [aixljdo  +  T,  [ablj/o  +  .    .  4-  r,»  [(m],jr'  +  t,^  [ab],y' 

+  .  .  =  r,  [an],      (5) 

We  t'oiicliide  that  the  piirtial  normal  equations,  when  the  full 
number  of  unknown  quantities  is  included,  may  be  derived 
from  those  of  the  form  (1)  by  the  following  rules. 

(1)  Each  partial  normal  equation  in  j?o,  yo,  .  .  .  is  formed 
from  that  in  x,  y,  etc.,  by  adjoining  to  the  first  member  of  the 
equation  the  member  itself  multiplied  by  r  and  then  changing 
X,  y,  .  .  .to  Xo,  Xu',  and,  in  the  products  by  r,  changing 
^,  y,    .     .    .    into  x',  y',    .    .    . 

(2)  The  partial  normal  equation  in  x',  y',  .  .  .  is  formed 
from  the  partial  equation  in  Xf>,  y^,  .  .  .  by  multiplying  all 
the  terms  throughout  by  the  factxjr  t. 

The  final  or  complete  normal  equations  in  all  the  unknown 
quantities  being  formed  by  the  addition  of  the  partial  normals, 
the  formula}  for  the  coefficients  are  as  follow : 


(2) 


(3) 


For  the  final  equation  in  Xq 

[aa]  =      [««],+       [aa]i  + 

[ab]  =      [a6J,  4-       [ab]i  + 
•    •  .    .  .    . 
[aa]'  =  n  [fl«J,  +  T2  [aa]2  + 

[an]    =       [an]i+      [«»]« + 

For  the  final  equation  in  x' 

[aa]"  =  Ti^aa\i  +  Ti'[aa\2  + 
[ab]"  =  ji^[ab]i+T2^ab]2  + 


•         •        • 


•        •        • 


«         •         • 


t         t         ■ 


•         t        • 


+  T„  [aa]„ 

•     •     t 

+       [«w]„ 


4-  T„*  [««]„ 


[an]"  =  r,  [an]i  +  tj  [an]^  +     .     .     .     +  r„  [an],, 


(6) 


(7) 


The  final  equations  for  all  the  unknown  quantities  will  then 
be  of  the  form 


[aa]  Xo  +  [ab]  y„  -f     .    .     +  [aa]'  x' +  ,    .    .  =  [an] 

•     •     •  •     •     •  •     •     • 

•     "     •  •     •     • 

[aa]'xo+[abyyo+  .    .    .   +[aa]"x'+  .    .    .  =[an]" 


(8) 


M 


52 


MEUCUUY,  VENUS,   AND  MARS. 


[25, 26 


Tlio  epoch  from  which  we  count  the  time,  r,  is  arbitrary. 
An  obvious  sulvantago  will  be  {jfaiued  in  countiuf^  it  from  the 
mid-epocli  of  all  the  observations.  Then  we  shall  have,  by 
putting  K'l,  Wt,  etc.,  for  the  sum  of  the  wei{,'lit8  for  the  different 
periods: 


M'l  Ti  +  Wi  Ti  + 


+  '»»  r„  =  0 


(9) 


If  the  observations  are  then  equally  distributed  around  the 
orbits  of  the  planet  and  of  the  Earth  it  may  be  expected  that 
the  coetlicients 


[aa\',  [ah]' 


(10) 


will  all  nearly  or  quite  vanish.  Practically  we  may  expect  that 
as  observations  are  continued  through  successive  rev^olutions 
the  ratios  of  these  to  the  other  coetlicients  will  approach  zero 
as  a  limit.  We  may  then  divide  the  normal  equations  into  two 
sets,  one  containing  the  ((uantities  .<■„,  ^d,  etc.,  and  the  other 
.r',  y',  etc.  The  coetlicients  (10)  being  small,  the  two  sets  of 
normals  will  be  nearly  independent,  and  we  may  omit  the 
terms  (10)  in  the  flrst  approximation,  and  introduce  them  in 
one  or  two  successive  approximations  so  far  as  necessary. 

The  unit  of  time  is  also  arbitrary.  A  certain  advantage  in 
synunetry  will  be  gained  by  so  choosing  it  that  the  mean  value 
of  T^  shall  not  differ  greatly  from  unity.  It  was  found  that 
twenty-tive  years  was  a  sutllciently  near  approximation  to  be 
adopted  for  all  three  planets. 

Dates  and  weights  for  epocha  and  periods. 

20.  As  want  of  space  m.akes  impracticable  the  present  publi- 
cation of  the  great  mass  of  material  worked  up,  the  following 
particulars  have  been  selected  as  those  most  likely  to  be  use- 
ful in  Judging  and  criticising  the  work.  We  give  three  tables, 
showing  the  division  of  the  dates  of  observation  into  periods, 
and  the  weights  for  each  period.  The  first  column  of  each 
table  contains  the  number  or  designation  of  the  period,  as 
found  in  the  manuscript  books.  The  second  contains  the 
mean  year  of  the  period.    The  third  column  shows  the  time 


I 


i 


-i 
« 


26] 


DATES  AND  WEIGHTS  FOR  EPOCHS  AND  PERIODS. 


53 


of  this  ineaii  period  from  the  mid-ei»och  of  the  ohservationis, 
which  is  takeu  us  follows: 

For  Mercury,  18<i5.0 
Vemia,  1803.0 
Mars,        18,5(1.0 

The  next  column  contains  the  snni  of  the  weights  of  the 
equations  in  each  period,  as  used  in  forming  the  normal  oipia 
tions.  These  were  not,  however,  tlie  woiglits  actually  used 
in  multiplying  the  coefllcients  of  the  equations  of  condition. 
Owing  to  tlie  diversity  in  the  quality  of  the  observations  at 
different  times  it  was  not  found  convenient  to  reduce  the 
equations  at  once  to  a  uniform  system  of  weights,  and  so  dif- 
ferent units  of  weight  were  selected  for  the  older  observations 
and  for  the  earlier  observations.  After  the  partial  normal 
equations  were  formed  they  were  multiplied  by  the  factor  F, 
necessary  to  reduce  them  to  a  standard  in  which  the  unit  of 
weight  should  correspond  to  the  mean  error — 

f,=  i  1".0 


The  sums  of  the  weights  reduced  by  these  factors  are  show^n 
In  the  table. 

In  arranging  the  weights  and  selecting  the  factors  it  should 
be  remarked  that  a  liberal  allowance  was  niade  at  each  step 
for  i)robable  constant  errors,  which  results  in  the  given 
weights  being  much  smaller  than  they  would  have  been  by 
the  theoretical  treatment  of  the  original  observations.  Not- 
withstanding this  allowance  the  final  result  seems  to  show 
that  it  was  still  insutlicient,  and  that  the  actual  weights  of 
the  results  are  less  than  would  follow  even  from  the  final  ones 
as  given. 

The  partial  normal  equations  for  each  period  after  being 
multiplied  by  the  factors  F,  are  added  to  form  the  final  normal 
equations  as  derived  from  meridian  observations. 


in 


< ) 


M  .">«  iRCURY,  VENUS,  AND  MA  US.  [26 

WeiijhtH,  epochH,  tnul  periods  of  partial  iiornud  cquntiom. 

MERCURY. 


Period. 

Right  Ascension. 

Declination. 

V. 

Mean 
year. 

1766.60 
1784.  22 
1799.81 

^                 VVt 
(units of  25;/.)     "  ' 

V. 
i 

Mean 
year. 

r 
(unitsr)r25.t.) 

Wt. 

I 
2 

-3-  93(io 
-3- 2312 
—2.6076 

3-4 
18.8 
26.1 

1765.50 
1782.99 

—3. 98o<:> 
—3.  2804 

0.  2 
4.9 

3i 

1796.4a 
1802.37 
1809.  18 
1824.  83 

-2.  7432 
—2.5052 
-2.  2328 
—1.6068 

5.0 

'x 

39-  9      ■?,! 
52. 8      ^a^ 

74-  I      ^^ 

1809.  53 

-2.  21S8 

18.9 

i 

1818.79 
1825.80 
i«?S-S6 

-1.8484 

—  I.  5'>8o 

—  I.  1776 

0.9 

34.5 
75.  0 

i 
i 

6 
6, 

V833;¥4 
1838.26 

•843-97 
1855.92 
1862.  79 
1867.  18 
1872.64 
1877.05 
18S2.  17 
1886.1:9 
l8oi;.  70 

—  1 .  2464 

—  1 . 0696 
—0.841..: 
-0.  3632 
—0. 0884 
+0. 0872 
+0.  3056 

1  0.4820 
+0.  6868 
-1^0.8516 
+0.  9880 

._. 

75-  ^      ^^ 

6, 

141.5 
281.5 
201. 5 
189.5 

294. : 
214.0 

204. 5 

«7«-5 

338.  0 
176.0 

7 
8 

9i 

9.' 
lo, 
lo. 
II, 
lij 
lij  1 

i 

1843. 74 

1855.90 
1863.  10 
1867.  12 
1872.62 
1877.12 
1882.24 
1886.29 
1889.  82 

—0.  8504 
—0.  3640 
—0.  0760 
+0.  0848 

4-0. 3048 

fo.  4848 
1  0. 6896 
+0.8516 

-j-o.  9928 

98.8 

83.3 
99.8 
186.0 
129.8 
129.8 
108.2 
199.8 
109.5 

i 
i 
i 

t\ 

i 
i 
i 
i 

VKNUS. 


I 

'7SS-83 

—4.  2868 

"•3 

* 

1759.  69 

-4.  1324 

7.0 

2 

1767.92 

—3.8032 

19.7 

* 

1770.  18 

-3- 7 '28 

10.0 

3 

1781.06 

-3-  2776 

3-7 

* 

1793.25 

—  2.  7900 

•3-5 

4 

1792.47 

—2.8212 

»2.  3 

* 

1S06.73 

—2.  2508 

65-5 

5 

1802. 64 

-2.4144 

23- 3 

* 

'815- 59 

—  1.8964 

67-5 

6 

1810.31 

—2.  1076 

34-0 

+ 

•823.75 

-1.5700 

197.0 

7 

1816.88 

-1.8448 

42.7 

* 

1836.02 

—  1.0792 

762.0 

8 

'825.55 

—I.  4980 

141. 0 

i 

1844.08 

-0.  7568 

650.0 

9 

'835.3I 

—  !.  1076 

339-3 

* 

1854. 24 

—0.  3504 

333-  0 

10 

1843-98 

— 0.  7608 

259-3 

* 

1861.43 

—0. 0628 

749.0 

II 

1853-51 

-0.  3796 

205-3 

* 

1868.06 

+0.  2024 

815.0 

12 

1861.60 

—0.  0560 

353-  7 

* 

•875-32 

-j-o.  4928 

692.0 

13 

1868.  12 

+0.  2048 

466.0 

+ 

1883.  15 

+0.  8060 

819.  0 

«4 

1875.  38 

+0.  4952 

399-  5 

i 

1888.56 

+  1.0224 

801.0 

15 

1883.  09 

4-0. 8016 

5'4-5 
520.5 

4 

16 

1888.  67 

+1.0268 

i 





m 


2«»,li71  UNKNOWN  QUANTITIES  OF  EC^UATIONH. 

W'eif/litti,  cporhH,  omi  periods  of  partial  not  >.ud  ei/initionM. 

MARS. 


05 


Kiijht  Asci-nsion. 

1 

! 
1 

1 

Menu 

r 

Wt. 

F. 

■« 

year. 

(un  tsdf  25  1. 

I 

'7S7.43 

—J.  942S 

•!53 

li 

2 

'770.55 

—J,  4180 

II.  0 

3 

17S7.82 

-2.7272 

U'   0 

if 

4 

•7W.  77 

—  2.  2492 

20.7 

5     ■?<>«.  32 

-1.7872 

•47 

6 

1829.  17 

-I.07J2 

60.  0 

7 

I8J7-39 

-0.  7444 

121.  0 

8 

1845-  39 

-0.  4244 

7t'-  3 

9 

1853.  36 

—0.  1056 

90.0 

lO 

1861.07 

-jo.  2028 

114.  u 

II 

1869.  20 

4-0. 5280 

124.0 

I  i 

12 

1877.71 

-f  0. 8684 

IJ2.0 

'3 

iSSjj  27 

4  1. 0908 

91.0 

14 

1S8S.8S 

^  1.3 140 

HS-S 

l>(.'clinntion. 


Mean  r  ' 

yrnr.     'lunilsnf  25  r  1 


Wt. 


1758.82 

'773-79 
1794.48 
|8<.>4.  91 
181  ].  00 
1828.04 
1 8,57.  18 
1S44.95 
1853.02 
|8()().  94 
1868.  80 
1877.38 
1 88}.  26 
1 888. 48 


-3-  8872 
3.  2884 

—  2.  4608 

-  -2.043() 

I,  7200 

1. 1184 

o.  752'^ 

o.  4420 

-o.  1192 

fo.  1976 

f  o.  5120 

+0.  8552 

41.  0904 

-'I.  2992 


8.8 

i 

8.8 

i 

13.0 

i 

47- 0 

i 

i    305 

i 

t    93.0 

37'" 

I 

255.0 

245.0 

306. 0 

197.0 

257- 0 

1 60. 0 

if)7.o 

I'lihioirn  tjuantities  0/  the  eqiidfionn. 

27.  For  convenience  in  solviii},'  the  eqnations  of  I'oiulitiou 
the  coenic'-Mit.s  of  tlie  ('qnation.s  were  innltiplicd  hy  .snch 
nnmerical  factors  a.s  wonhl  rednce  tlicir  jfcneral  mean  ab.so- 
Uite  value  to  nnmoers  of  a|)pro.\iiiMit  1y  the  .same  order  of 
niagnitnde.  Hence,  the  unknown  <iuantities  themselves  are 
not  the  corrections  to  the  elements,  but  the.se  conections 
divide<l  by  the  adopted  factors. 

In  the  ea.se  of  3Iercury  tlie  absolute  term  was  also  multi- 
jdied  by  10,  so  that  eUcctively  the  factor.s  in  question  were 
reduced  to  one-tenth  part  of  their  vahie.  Tiie  unknown 
quantities  of  the  equa.tions  arc  rejiresented  by  tiie  symbols 
of  the  elements  to  wliich  they  relate  inclose<l  in  brackets. 

For  convenience  of  reference  the  followiu};-  table  is  {jiven, 
showing  the  factors  used  i'l  the  ca.se  of  each  planet.  In  the 
ea.se  of  Mercury  the  column  (a)  shows  the  factors  by  which  the 
difterential  coefticients  were  actually  multiplied :  (/>)  the  factor 
by  which  the  unknown  (juantity,  as  tinally  found,  must  be 


tk^ 


MERCURY,  VENUS,  AND  MArfS. 


[27,  28 


iMultipIied  to  obtain  the  correction  as  expressed  iu  the  last 
column.  In  the  case  of  Venus  and  Mars  these  factors  are  the 
same. 

Factor»  by  which  the  unknotcn  quantitiefi  are  to  be  multiplied  to 
obtain  corrections  of  the  elements. 


Symbol  of 

Factor  for — 

Corr.  of 

unknown. 

Mercury 

(") 

(/') 

Venus. 

Mars. 

element. 

WJ 

1 

0.1 

7 

0.3 

6m  :  Wo 

/     ] 

40 

4 

5 

2 

6\ 

I    J] 

30 

3 

G 

2.5 

tfj 

|NJ 

30 

3 

7 

2.5 

sin  JfJN 

1   ^'  1 

30 

3 

3 

10^7 

8e 

Tt 

100 

10 

439 

100^7 

dn 

Tt 

100 

2.05G 

3 

1.3323 

edn 

^      1 

10 

1 

4 

4 

dt 

.   «" 

6 

0.6 

2.5 

2 

6e" 

n" 

6 

0.0 

2 

2 

e"6n" 

[   <^  \ 

10 

1 

1 

5 

a 

^  \ 

10 

1 

5 

5 

S 

I"  \ 

10 

1 

4 

3 

61" 

The  secular  variation  of  each  unknown  in  2~)  years  is 
e.tpressed  sometimes  by  a  suttixed  1,  sometimes  by  an  accent, 
thus: 

[/ 1'  =  |/],  =  change  of  [/]  in  2.j  years. 

28.  It  may  also  oe  useful  to  give  the  values  of  the  principal 
coenicients  in  each  of  the  normal  equations.  They  are  found 
in  tlie  followinj;  table.  Were  the  other  coefficients  all  zero, 
tliose  numbers  would  inJicate  the  weights  of  the  different 
unknown  nuantities  as  resulting  from  the  solution.  Several 
of  them  were  greatly  diminished  by  the  i)roce8S  of  solution. 


28,29] 


ORDER  OP  ELIMINATION. 


5T 


Values    of   the  prineipai  diagonal  coefficienU  in  the  normal 

equations. 


Mercury. 

Venus. 

Mars. 

Symbol  of 

From 

coefficient. 

From  mer. 
observa- 

From 
transits. 

Sum. 

1"  rom  mer. 
observa- 

From 

transits. 

.Surn. 

mer. 
observa- 

tions. 

tions. 

tions. 

[  '"•'  ] 

5488 

0 

5488 

5868 

2929 

8797 

17887 

^  // 

10559 

1 1 308 

21867 

5981 

3540 

9521 

20924 

r  n  1 

15222 

1296 

16518 

13232 

7444 

20676 

28783 

-  XN  ■ 

14176 

2304 

16480 

1 795 1 

1636 

19587 

32478 

e'  t' 

19015 

5076 

24091 

5686 

3350 

9036 

201 19 

TT  TV 

8621 

8352 

'6973 

5290 

1732 

7022 

20564 

f  e 

IIOOI 

196 

11197 

1 1429 

3598 

15027 

31460 

■,,//,.//■ 

9757 

508 

10265 

9586 

665 

1025 1 

15909 

\"  tt"' 

9099 

261 

9360 

5836 

1895 

7731 

14911 

,if  ^>t  ■ 

5242 
1 304 1 

0 

5242 
13583 

'  l"l"  ' 

542 

11031 

2349 

53380 

15427 

na 

13230 

0 

13230 

.335 

0 

335 

25138 

■  riJ   ■ 

24657 

0 

24657 

15196 

0 

15196 

53975 

'   //   ' 

/ 

7014 

67155 

74169 

6005 

8983 

14988 

26689 

n 

/ 

12366 

9383 

21749 

9837 

13014 

228';i 

23440 

XN  ■ 

/ 

"035 

16682 

27717 

14724 

2874 

17598 

29494 

('  (* 

/ 

15437 

29647 

45084 

5743 

8610 

14353 

24364 

ff  77 

/ 

6745 

493 « 8 

56063 

4948 

4483 

9431 

27131 

f,  f 

/ 

8488 

1418 

9906 

8458 

6306 

14764 

25675 

'  ,•''  e'^  ' 

/ 

8409 

29o; 

1 1346 

9805 

1682 

11487 

22947 

tt"  tt'' 

/ 

8410 

I5»3 

9952 

1    5242 

4805 

10047 

17356 

■ ," r"^ 

/ 

54)2 

0 

S432 

,  I"  I"  ' 

/ 

1 1629 

3126 

•4755 

10677 

5667 

16344 

2065s 

a  (I 

/ 

1 1400 

0 

1 1400 

i    297 

0 

297 

33624 

[  ''''  '. 

/ 

18716 

0 

18716 

10772 

0 

10772 

42405 

NoTK. — The  coefficients  for  Mercury  and  Venn?  in  this  table  are  j^iven  as  they 
were  used  in  the  solution,  after  droppinii;  the  units  from  all  tiie  terms  of  the 
equations,  except  those  from  transits  of  Nieicury. 

Order  of  climinafion. 

29.  Ill  dealing  with  so  extonsive  a  system  of  uuknown 
quantities  it  is  impracticable  to  inveHti}jrate  tlio  depeiMlence  of 
Oiich  upon  all  the  others.  It  is  therefore  essentia)  to  arrange 
the  unknowns  in  an  order  partly  that  of  interdependence  and 
partly  that  of  the  liability  of  em'h  to  subse(|uent  change  by 
discussion  and  adjustment.  Hence,  tlie  mass  of  the  planet, 
Mercury  or  Venus,  should  be  first  eliminatad,  as  being  that 
unknown  which  is  least  affected  by  changes  in  the  tinal  values 
of  the  other  unknowns.    The  secular  variations,  as  derived 


M 


MERCUrY,  VENUS,  AND  MARS. 


[29,30 


!         N 


from  meridisiii  observations,  are  nearly  indei>endent  of  the 
corrections  to  tlio  other  elements.  The  solar  elements  are  to 
be  subswjuently  determined  by  a  combination  (►f  the  results 
of  the  observations  of  the  Sun  and  of  the  three  i)lanets. 

Guided  by  these  considerations,  the  order  of  elimination 
was,  with  some  exceptions,  as  follows: 

1.  The  mass  of  the  disturbing  planet. 

2.  The  live  elements  of  the  observed  planet. 

3.  Tlie  four  elements  of  the  Karth's  orbit. 

4.  The  corrections  to  the  star-positions  for  the  mid-epoch. 

5.  The  secular  variations  of  the  eleven  tpiantities  (2),  (3), 
and  (4),  taken  in  the  same  order. 

Treatment  of  meridian  ohservations  of  Mercury. 

30.  In  tlie  case  of  ]Mercury  the  factors  of  the  coetticients  of 
the  equations  were  chosen  large  enough  to  admit  of  the  deci- 
mals being  dropi>e<l  from  the  products  without  prejudice  to 
the  accuracy  of  the  final  result.  This  was  done  to  facilitate 
the  formation  of  the  normal  e<|uations.  For  the  same  reason 
the  factors  were  made  so  small  that  the  absolute  numerical 
values  of  the  coelflcii'nts  should  generally  not  exceed  13.  As 
tiiis  degree  of  precision  is  far  short  of  that  usually  emidoyed 
for  correcting  the  elements  of  a  i)lanet,  it  may  be  well  to  set 
forth  the  considerations  on  which  it  is  based. 

Let  any  equation  of  conditi(»n  as  actually  used  be — 


"•'•  +  %  4-  ^*  + 


=  n 


(a) 


Let  the  coetticients  a,  h,  etc.,  be  attected  by  the  mean  errors 
f,  f',  etc.,  so  that  the  true  equation  should  be — 

(a  +  f)j'  +  (&+^')i/+     .     .     .     =H. 

This  true  e(|uation  may  be  written  in  the  form — 

*w  •  +  %  +     .     .    .    =  n  —  ex  —  f'y  —     .    ,    .  (b) 

We  nuiy  regard  (h)  as  a  rigorous  eipiation,  in  which  the  error 
of  the  second  mend>er  is  increased  by  the  ((uautity — 


rt  f J?  ±  f'y  .-t . 


aoi 


MBUIDIAN  OBSERVATIONS  OP  MERCURY. 


s» 


and  tlie  only  effect  n|M)n  the  precision  of  the  results  will  be 
that  arising  from  this  increased  probable  error.  Let  us  esti- 
mate Its  magnitude.  From  an  examination  of  the  tabh'S  used 
in  iinding  the  coellicients  I  infer  that  the  probable  error  of  the 
eoet1i<'ient  of  n  was  I:  1,  and  tiiat  of  all  the  other  coetlicients 
rli  0.<».  The  mean  value  of  the  unknown  «|uantities  M-as  gener- 
ally a  small  fraction  of  a  second.  We  conclude,  therefore, 
that  the  probable  w  mean  value  of  the  «'rror 

A  t.r  \    >!f  \       ... 

would  in  any  casi^  be  only  a  small  frai'tion  of  a  second.  More- 
ov«*r,  these  errors  would  be  purely  accidental  and  not  syst«'n)- 
atic,  since  the  intervals  of  time  between  the  e(|uations  were 
generally  so  long  that  the  coefticicnts  for  ditlerent  equations 
came  from  «liti'erent  tables,  so  that  no  error  from  omitted  deci- 
mals in  anyone  e(|uation  would  enter  into  the  other  equations. 

Now,  in  view  of  the  necessary  systematic  errors  which  affect 
observations  of  the  planets,  there  is  no  hope  of  approxim?'~.iig 
to  this  degre<'  of  a«'(!urary  in  the  secoiul  members  of  the  <'qtni- 
ti(Mis.  Were  the  observations  rigorously  correc^t  and  the 
values  oftho  unknown  (|uantities  finally  determined  affe«'ted 
by  no  eiror  except  that  arising  in  this  way,  they  wjadd  be 
many  tinuvs  more  a<;curate  than  we  can  hope  to  make  tln-m. 
The  «'rrors  miglit,  in  fact,  be  considered  unimportant  in  tl>« 
present  state  <>f  astroufuny. 

It  has  alrea<ly  been  reiuarked  that  the  scale  of  weights  was 
80  taken  that  the  unit  of  weigiit  shouhl  <'orrespond  approx- 
imately to  a  sup|iosed  mean  eiror  I  l".0  in  the  value  <»f  each 
absolute  term  of  an  «M|uation  of  condition,  so  far  as  the  error 
could  he  det<u'miiied  from  the  discordance  of  the  tu'iginal 
observations.  The  corresponding  probable  error  would  be 
i  f)".(M.  In  the  case  of  Mercury,  however,  modili(;ati(Uis  were 
made  which  i>i'events  this  mean  err(U"  from  «'orrespon«ling  to 
the  unit  of  weigiit  which  would  be  found  from  the  solutions  in 
the  usual  way.  In  the  first  phu-e,  the  absolute  members  were 
id!  multiplied  by  !(►;  in  other  words,  the  decimal  point  was 
dropped  from  tenths  of  sccon«ls.  and  no  further  account  taken 
of  it.  Secon«lly,  in  (;onse(|ueuce  of  the  probable  error  in  the 
coefllcients  of  the  normal  equations  arising  from  the  imperfec- 


i 


60 


MKRCUKY,  VENUS,  AND  MARS. 


tioiis  of  the  decimals,  the  final  values  of  these  coeiTicients 
would  be  subject  to  probable  errors  ranging  between  50  aud 
ICO  units.  In  consequence  there  would  be  no  advantage  in 
retaining  the  last  figure  in  the  normal  equations,  and  it  was 
dropped  in  all  the  subsequent  solution  and  discussion  of  these 
equations. 

In  dropping  the  last  figure  from  the  absolute  term  of  the 
normal  equations  we  may  consider  that  we  are  merely  drop- 
ping the  tenths  of  seconds  and  that  the  units  are  once  more 
expressed  in  seconds.  Tliu »,  considering  only  the  efli'ect  of 
this  oi)eration,  the  unit  of  weight  would  correspond  to  a  mean 
error  of  i  1.0  in  units  of  the  absolute  term.  But  in  dropping 
off  the  last  figure  from  the  coeflicients  we  i)ractically  reduce 
the  scale  of  weights,  considered  as  multipliers  of  the  equa- 
tions, to  one-tenth  of  their  former  value.  On  the  other  hand, 
in  expressing  the  unknown  quantities  in  terms  of  the  correc- 
tions to  the  elements,  we  divide  the  nmltipliers  by  ten,  so  that 
effectively  we  nmltiplied  the  coefficients  in  the  ei|uations  of 
condition,  considering  the  unknown  (quantities  to  be  defined 
as  on  page  56,  by  10.  Since  these  coefficients  are  of  the  second 
degree  in  the  normal  equations,  it  follows  that  the  scale  of 
weights  has  in  eftect  been  increased  ten  fold.  Hence  the  unit 
of  weight  for  the  normal  equations  between  the  unknown 
quantities  as  finally  solved  will  correspond  to  the  mean  error 

f,  =  1.0x  \/l0=i3.1 


As  the  mean  error  is  at  best  a  rather  indefinite  quantity  in  a 
case  like  the  present,  we  may  consider  its  value  as  4  uniif^:  and 
even  then  as  by  no  means  rigorously  determined. 

Up  to  the  time  of  writing  no  attempt  has  been  made  to 
derive  rigorously  the  weights  of  the  unknown  quantities  from 
the  solution,  because  in  the  cases  of  most  of  the  unkowns  such 
weights  would  be  entirely  illusory.  Tlie  fact  is  that  in  solving 
so  immense  a  nmss  of  equations,  we  must  expect  systematic 
errors  to  vitiate  many  of  the  results.  The  observations  of 
Mercury,  especially  of  its  Kight  Ascension,  are  not  nmde  on 
a  uniform  system;  sometimes  the  limb  is  observed,  sometimes 
the  apparent  center  or  the  center  of  light. 


30,31 


TRANSITS  OF  MEUCl'RY. 


Gl 


All  ideally  perfect  'ysteiii  of  reduction  would  require  us  to 
reduce  each  separate  observation  with  a  seniitlianu»ter  corre 
spondinj;  to  the  personal  equation  of  tlie  observer.  This  being 
entirely  impracticable,  we  nuist  reganl  the  reduction  of  the 
observer's  seinidianieter  to  that  used  in  tlie  reductions  as  a 
probable  error.  In  fact,  however,  it  will  be  of  a  systematic 
character,  varying  at  each  point  of  the  relative  orbit  of 
^fercury,  and  going  through  a  cycle  of  changes  impossible  to 
determine  in  ii  synodic  perioil  of  the  planet.  It  is  impracti- 
cable to  give  even  a  full  discussion  of  these  errors;  we  shall, 
however,  meet  with  a  proof  of  their  magnitude. 

Introduction  of  the  cqitations  (h'rireil  from  ohitcrecft  trouxits  of 

Merc  or  If. 

M.  The  relations  between  the  elements  of  Mercury  and  the 
Karth  tlerived  from  this  sour<;e  are  shown  in  my  Dincusniou  of 
Transits  of  Mercuri/  (A.  P..  Vol,  I,  Fart  VI.)  (hi  page  417  are 
found  e.\i)ressions  for  those  linear  functions  of  the  corret'tions 
to  the  elements  whi<;h  are  deteiinini'd  by  the  November  .md 
May  transits,  resjtectively.  With  a  slight  change  (►f  notation 
to  correspond  with  that  «>f  the  i)resent  paper,  these  functions 
are  as  follows: 

V  =  1.487  61  —  0.487  rtV  -  1.137  rfc  -  1.01  rU"  +  1.19  e"dn" 

+  l.uSfV 
\V  =  0.7U;  (U  +  O.L'84  r»  +  0.8«<J  6e  —  0.97  rtV"  -  1.11  c"(W' 

-  1.62  Se" 

The  values  of  V  and  W  being  derived  from  a  series  of  transits 
extending  from  1077  to  the  present  time,  enable  us  to  deter- 
mine both  these  quantities  at  some  epoch,  and  their  secular 
variations.  The  values  derived  from  tlie  transits,  together 
with  their  mean  errors,  are  found  on  page  4(»0  of  the  work  in 
question.  Omitting  the  d<»ubtful  factor  A*,  introduced  on 
account  of  a  i)os8ibie  variability  of  the  Karth's  axial  rotation, 
which  was  not  proved  by  the  transits,  the  values  of  V  and  W 
were  found  to  be  as  follows: 


V  =  —  0".90  i  0".31  +  {-'J".i\3  ±  0".50)  (T  -  1820) 
W  =  +  0".84  ±  0".25  +  (+  1".84  ±  0".G0)  (T  -  1820) 


(«) 


fi2 


MERCURY,  VENUS,   AND  MARS. 


[31 


' 


The  mean  ei><)ch  for  the  transits  is  taken  as  1820,  to  which 
the  zero  vahies  correspond.  The  valnes  for  18G5.0,  the  mid 
epoch  for  the  meridian  observations,  are,  therefore,  from  the 
transits  alone — 

V  =  -  L"'.08  ±  0".41 
W  =  +  1".«7  i  0".37 

This,  however,  is  only  a  tirst  approximation  to  the  quantities 
which  shouhl  he  introduced.  Since  the  meridian  observations 
help  to  determine  the  values  of  V  and  W,  we  should  not 
regard  the  reductions  to  1805.0  as  final,  but  retain  the  results 
in  the  form  (<»). 

Another  element  which  is  determined  from  the  observed 
transits  of  Mercury  with  greater  precision  than  it  can  be  from 
meridian  observations  is  the  longitude  of  the  node  of  the  orbit 
relatively  to  the  Sun.    In  the  paper  quoted  we  have  put — 


a!»d  found  from  all  the  transits  up  to  1881, 

N  =  -  0".1({  ±  0".27  +  (0".28  ±  0".r>2)  (T  -  1820) 


(fr) 


The  values  of  V,  W,  and  N,  found  from  the  discussion  in 
question,  give  rise  to  six  conditional  equations,  which  become 
completely  independent  when  we  take  as  observed  values  the 
secular  motions  and  the  absolute  valines  at  the  mid-epoch  of 
observation.  This  mid-epoch  is  not  the  same  for  the  May  and 
November  transits.  But  I  have  assumed  that  no  serious  error 
would  be  introduced  by  talcing  1820.0  as  the  epoch  for  all  three 
of  the  quantities,  V,  W,  and  N. 

If  we  substitute  for  sin  i  6ft  its  value  in  terms  of  6J,  etc., 
namely, 

Sin  i6f)=  -  O.fiOlS  6,1  +  0.7»«»  sin  JdN  -f  0.721  6e        (c) 

and  then  for  6 J,  ($N,  6fy  their  values  in  terms  of  the  unknowns 
of  the  equations  of  condition,  we  shall  have 

N  =  -  1.805  [J]  +  2.394  [N]  +  0.721  [e]  -  0.122  [I"]      {d) 


311 


TRANSITS  OF  MEBCl  RY. 


63 


Similar  expressions  will  be  found  for  the  values  (»f  V  and  W 
by  substituting  for  tlie  corrections  to  tlie  elements  tlie  unknown 
quantities  of  the  conditional  equations,  as  already  given. 

Taking  181*0.0  as  the  mid  epoch,  we  may  regard  the  inde- 
pendent quantities  given  by  the  transittj  of  Mercury  to  be  the 
six  following  ones : 


Vn 


1.8  V 
V, 


W„-!.8\V,:  X„-  1.8  N, 


W 


{e) 


Here  Vn,  Wq,  and  N,,  indicate  values  for  1865,  the  mid-epoch  of 
the  meridian  ob8ervati<ms;  and  V,,  Wi,  an«l  Xi  the  variations 
in  :;.">  years.  The  six  conditional  equations  thus  found  from 
the  transits  may  be  written 


Vo 

-1.8  V, 

Wo 

-1.8W, 

No 

-  1.8  N, 

V, 

w, 

N, 

-  0".9O  1:  0".31 

+  0".s4  -J:  o".i>ri 

-  0".l<i  ]  0".27 
~0"M:L  0".lu 

+  0".4<;ri:  0".15 

-h  o".o7 1  o'.ir) 


(/) 


Substituting  for  Vn,  Vi,  etc.,  their  expressions  as  linear  func- 
tions of  the  unknowns  of  the  conditional  equations,  we  find 
the  following  six  equatioits,  which  are  to  be  used  as  conditioiml 
equations  additional  to  those  given  by  the  meridian  observa- 
tions : 

5.9.-)  [I]  -  4.87  [7t\  -  3.41  [e\  -  1.01  [l"\  -}-  0.71  \rr"\  +  0.05  [e"] 
-1.8!5.95[/],-4.87[;r],-3.41[el,-1.01[/"J,+  0.7l[;r"j, 
-f  0.95[e"J,|  = -0".90 

Weight  =  260 

2.80  [/]  +  2.84  [tt]  +  2.09  [e]  -  0.97  [l"\  -  0.07  \7i"\  -  0.07  [e"] 
-1.8|2.80[/j,+2.84(7r],  +  2.09[e], -0.97[/"J,-0.07[;r"J, 
-0.97[c"],|  =  +0".84 

Weight  =  300 

-  1.8  [J]  4-  2.4  [N]  -f  0."  [e]  -  0.12  [/"] 
- 1.8  {  -  1.8  [J],  +  2.4  [NJ,  +  0.7  [f  ],  -  0.12  [/"], ;  =  -  0".16 

Weight  =  400 


64 


MERCURY,  VKNUS,  AND  MARS. 


[31 


5.05  [/],  -  4.87  [n-],  -  3.41  [e\i  -  1.01  [/"],  -f  0.71  [t"],+  0.05  |e"], 
=  -  0".«« 

Weight  =  700 


2.80  [/],-!-  2.84  \7r]i  +  2.00  [c],  -  0.07  [/"],  -  0.07  [;r"],  -  0.07  fe"], 
=  +  0".40 

Weight  =  700 


-1.8  [J],  +  2.4  [N],  4-  0.7  [f]i  -  0.12  [/"],  =  +  0".07 
Weight  =  1,000 


The  weights  assigned  to  tliese  several  equations  have  been 
determine*!  by  the  following  considerations: 

We  have  ahead"  found  that  in  the  equations  of  condition 
from  the  meridian  observations  as  tinally  reduced,  the  scale  of 
weights  has  so  come  out  as  to  show  a  practical  mean  error  for 
weight  unity  of  about  i  4".  Were  this  error  jmrely  accidental, 
the  weights  of  the  conditional  equations  derived  from  the 
transits  would  be  determined  in  the  same  way,  from  the  mean 
errors  assigned  to  them.  But,  as  a  matter  of  iiwjt,  the  exist- 
ence of  systenuttic  erroi'S  in  the  meridian  observations  is 
shown,  as  will  be  subse<iuently  explained,  by  the  large  value 
found  for  the  tictitious  quantity  rf/'j.  Since  observations  of 
transits  are  made  at  the  point  of  the.  relative  orbits  of  Mercury 
and  the  Karth,  near  which  meridian  observations  are  rarely 
available,  and  are  of  a  higher  order  >f  accursicy  than  meridian 
observations,  it  follows  from  the  theory  of  probabilities  that 
we  should  assign  a  larger  relative  weight  to  the  observations 
of  the  transits.  How  much  larger  does  not  admit  of  being 
determined  with  numerical  precision.  Actually  I  have  taken 
the  weights  as  if  the  mean  error  corresponding  to  weight 
unity  were  between  5  and  <>.  In  the  case  of  the  motion  of  the 
node  a  still  larger  weight  has  been  assigned  to  the  secular 
variation,  from  the  belief  that  the  accuracy  of  the  determina- 
tion from  transits  relative  to  meridian  observations  is  in  this 
case  of  a  yet  higher  order  of  magnitude  than  in  the  case  of 


31,  32|        SOLUTION  OF  ElilATIONS  FOU   MKUCl'RY.  60 

tli«  other  oh'meiits.     Whether  this  belief  is  jnstiHe<l  or  not 
must  be  left  to  tlie  decision  of  the  future  astronomer. 

The  tirst  three  of  the  preceding?  six  eonditional  equations 
may  be  treated  in  a  way  simihir  to  that  a(h»ptetl  for  tl>e 
meridian  observations.  They  express  what  is  supposed  to  [ij 
equivalent  to  observations  of  the  tiiree  quantities  V,  VV,  and 
N  in  18L'0,  when  r  =  —  1.8.  Ilenee,  fnun  the  partial  nornuils 
in  the  si's  lu-ini'ipal  unknowns,  [f],  [t-]  .  .  .  [r"],  the  eom- 
plete  normals  nuiy  be  formed  by  nudtiplication  by  r  and  7* 
(r  =  —  1.8)  in  the  way  set  lorth  in  §2.">. 

SolittioitH  of  the  equations/or  Mcrcurif. 

^2.  In  the  case  of  Mercury  ami  Venus,  it  is  desirable  to 
know  to  what  extent  the  results  of  the  transits  diver^je  from 
those  of  the  meridian  observations.  Hence,  as  already 
remarked,  two  solutions  of  the  equations  were  made,  termed 
A  and  li. 

Solution  A  is  that  derived  from  the  meridian  observations 
alone.  Solution  B  is  that  of  the  nornnil  equations  formed 
from  both  the  meridian  observati<ms  and  the  transits. 

The  results  of  the  solutions  in  the  case  of  Mercury  are  shown 
in  the  followiufj  tables.  The  relation  of  the  unknown  quan- 
tities {jfiven  in  the  tirst  columns,  A  and  H,  to  the  corrections 
of  the  elements  has  been  shown  in  a  preceding?  section  (§  27). 
The  upper  half  of  the  table  shows  the  corrections  to  the 
elements;  the  lower  half  those  of  the  secular  variations. 

It  will  be  seen  that  all  the  values,  with  a  single  exception, 
c«»me  out  less  than  a  unit.  In  stating  the  corrections  to  the 
elements,  it  must  be  remembered  that,  owing  to  the  proximity 
of  Mercury  to  the  Sun,  the  errors  of  geocentric  plai-e  are  much 
less  than  those  of  the  heliocentric  elements,  so  that  an  error 
in  the  hitter  indicates  a  proportionally  smaller  error  in  the 
actiuil  observations.  For  the  same  reason  we  nuist  expect  a 
less  degree  of  precision  in  the  elements  as  finally  derived  than 
in  the  case  of  the  other  planets. 
5G90  N  ALM 5 


06 


'ii    k 


MEBCUBY,  YENUM,  AMU  MABS 

MF.RCL'KV. 

RvHultx  it/  HolntUniH  of  the  normal  equations. 


[32,  33 


Unknowns. 

< 

(  orrections  of  clcmenti. 

Syml)ol. 

A. 

H. 

.Symbol. 

1 

A. 

n. 

['"'] 

— o.  1478 

-0.  1207 

0. 1 

ft  III :  /// 

—0.0148 
// 

—0.0121 

/  " 

-0.  1342 

-0.0752 

4- 

.1/ 

-0.  537 

—0. 301 

■  .1  ■ 

-0.  2436 

—0.  2;!99 

3- 

''J 

-0.  731 

—0. 690 

;n 

—0.  0227 

—0.  OJOI 

3- 

Sin  1  rl  \ 

0.068 

—0.061 

t 

f  0.  2074 

1  0.  2194 

1. 

<ff 

1  0.  207 

f  0. 219 

i' 

—0.  1202 

f  0.  4094 

3- 

ie 

—0.361 

4 1. 228 

TT 

t  0.  5209 

+0.  2688 

10. 

.U 

+5.209 

f2.689 

'.-" 

-f  0.  0669 

-j-o.  8397 

0.0 

<1 , " 

to.  040. 

+0.504 

'"■''j 

—0.  2248 

—0.  7027 

0.6 

,.// ,( ff// 

-0.135 
+2.248 

—0.  422 

',.//■ 

4-1.  1240 

+  1.0566 

2. 

.J  / " 

f2.  113 

'    A    ' 

~o.  2310 

—0.  2556 

I. 

<1 

—0.  231 

-—0.  256 

'  l"\ 

-0. 0354 

—0.  0897 

1. 

tM" 

-0. 035 

—0.090 

a  \ 

+0. 4803 

1  0.  493«5 

I- 

a 

-\-o,  480 

i  0.  493 

I  ■ 

/ 

— 0.  20(J0 

—0.  1209 

16. 

I),  <V 

—3-  296 

— '^.'S 

:j  : 

' 

— 0.0114 

f  0.  0636 

12. 

1),,)J 

-0.  137 

+  0.  764 

_n; 

/ 

-j-o.  1000 

f-o.  0930 

13. 

SinJI),<lN 

1  I.  200 

+  1.  116 

t  \ 

/ 

4-0.  0681 

-f  0. 0966 

4- 

D.cJe 

+0.  272 

+0.  386 

t* 

' 

—0,  1165 

+  0.0987 

12. 

I),.),- 

-I.  398 

+  1.  184 

:r 

/ 

-0.  2385 

—0.  0252 

40. 

",<5t 

—9.  540 

—  1.008 

\.n' 

/ 

—0.  1968 

+0.  1317 

2.4 

1  ,&e" 

—0.  472 

1  0.  316 

'n" 

/ 

—0.  1677 

—0.  1 193 

2.4 

i'"  I ),  (S  v" 

—0.  402 

-0.  286 

'r" 

/ 

|-o.  1 108 

+0.0806 

8. 

\\i\r" 

+0.  886 

+0.645 

'  d  ' 

/ 

—0. 1826 

-0.  1233 

4. 

D,d 

-0.  730 

-0. 493 

\'"\ 

/ 

— 0.  1442 

-0.3152 

4- 

V>^dl" 

-0.  577 

—  I.  261 

\  a  \ 

/ 

—0.  3160 

-o- 1973 

4- 

D,a 

—  I.  264 

-0.789 

Mean  ejioch  of  corrections,  1865.0. 

DiHCortJance  in  the  ohnerred  Right  Ancennions  of  Mercnrti. 

33.  The  most  remarkable  feature  in  tlie  result  i.s  the  value 
of  the  (piantity  represented  by  [/•"].  The  unkiiowu  quantity 
introduced  with  this  symbol  had  as  its  roetticient  the  derivative 
of  the  geocentric  phu^e  as  to  the  Earth's  rmlius  vector,  and  the 
result  would  therefore  be  an  apparent  constant  correction  to 
that  radius  vector.  Since,  however,  the  position  of  the  planet 
depends  (mly  on  the  ratio  of  the  distances  of  the  I<]arth  and 
Mercury,  it  follows  that  the  actual  correction  may  be  regarded 
as  a  correction  to  the  ratio  of  the  mean  distances. 

The  determination  of  the  mean  distances  by  Kepler's 
third  law  may  be  regarded  as  so  unquestionable  that  the  true 


'^1 


^". 


,  r'.WwHK-  ■  ttMWg»WW»M>UjsJ^^1M> 


33) 


DISCOllDANCK   OF   (»nKERVATI()N8, 


67 


value  of  tliis  unknown  «|uantity  hIiouIiI  lu'  n^i^ardcd  an  /oro, 
aiul  tliu  result  as  a  purely  tictitious  one,  arising  from  errone- 
ous eluiui'iits  of  reduction  or  systematic'  personal  errors.  It 
was  the  possibility  of  the  latter  that  led  to  its  introduetion. 
When  the  planet  is  i^ast  of  the  Sun,  observations  are  always 
nnide  on  or  near  its  west  limb,  or  at  h'ast  on  some  point  west 
of  tlie  true  tenter,  and  vice  rcrna.  The  value  <»f  <W'  therefore 
indicates  tliat  there  is  a  lemarkable  systematic  dill'erenee  in 
the  observed  Kij^ht  Ascension  aircordinj;-  as  the  phuu't  is  east 
(u  west  of  the  Hun,  and  therefore  according  to  the  illuminated 
side.  The  sifjii  of  the  result  sh<»ws  that  the  reduction  to  the 
center  of  the  ])lanct  was  apparently  too  small.  It  is  there- 
lore  of  interest  to  learn  according  to  what  law  this  error 
changed  as  the  planet  nn)ved  around  its  relative  orbit. 

It  has  up  to  the  present  time  been  impracticable  to  substi- 
tute the  unknown  quantities  in  the  original  e<pnitions  of  con- 
dition, ami  thus  determine  the  separate  resijbmivij  and  for  the 
](urpose  of  investigating  the  present  case  such  a  substitution 
is  the  less  necessary,  owing  to  the  smallness  of  the  unknown 
(|uantities.  I  have  therefore  8ini|)ly  determined  the  mean 
correction  lo  the  Higlit  Ascension  given  by  all  the  oi)serva- 
tions  during  the  various  periods  in  six  segments  of  the  i-elative 
orbit,  near  the  elongations,  and  before  and  after  the  two 
conjunctions.  The  residts  are  shown  in  the  following  table. 
Commencing  with  the  moment  of  inferior  conjunction,  column 
A  contains  the  mean  correction  to  the  tabular  Right  Astcnsion, 
from  observations  made  within  about  twenty  days  following. 
Column  H  contains  the  observations  made  from  twenty  days 
af'f'rthe  inferior  conjunction  until  twenty  days  bef'  re  superior 
con,juncti(»n,  a  period  during  whi(di  the  planet  w.is  generally 
near  its  greatest  west  elongati(»n.  Column  C  contains  the 
observations  made  dining  the  twenty  days  foHowing  and  up 
to  superior  conjunction.  Then  follow^  in  regular  order  the 
corresponding  results  when  the  planet  was  east  of  the  Suu, 
beginning  with  the  twenty  days  fidlowing  superior  conjunc- 
tion and  going  around  to  inferior  conjunction. 


<M 


MUUCrUY,  VBNim,  AND  MAUM. 


133 


^ 


I  ' 


Table  Hhoirintf  the  mean  annrtionH  to  the  tohular  Uight  AnceU' 
ninn  of  Mervtiri/  in  nix  HvijiiuntH  of  Hh  irlutire  orbit. 


KpocliH, 

A 

n 

f      lot. 

t  2.  61      5 

.1.83     10 
-fi.79    24 
HI. 48     18 
t  0.  77     20 
1.14    72 
1  0.  74    65 
-f-0.98    63 

C 

''        w/. 

fo.97      4 
f«i3     24 
— 0.  38     20 
t-o.  08     16 
fo.  31     44 
-0.  20    61 
—0.  15     62 

'7''S-I79« 

1793-181S 

1817-1839 

1840-1849 

1850-1859 

1860-1869 

1870-1880 

1881-1892 

f        wt. 

f  3  24      4 

f  2.  06      6 
\  \.  06       6 
f-l..»6       6 

f3-  72       4 
-fl.18     28 

ti.l8     25 

-fl.19     .^8 

D 

E 

F 

>7'>5-'79l 

1793-1X15 

1817-1839 

l8.<o-i849 

185^-1859 

l86)-l869 

i87o-i:;;kj. 

1881-1892 

''        wt. 
+0.92      1.5 
+2.82      5 
J-o.  27     25 

to.  22      22 

-f  0.  69       14 
-0.  44      55 

-0.52     57 

-0.84      80 

-fl.30     10 
-f  I.  10     16 
43.76     24 

-0.53     30 
-0. 39     28 
-0.55     69 
-I.  25      67 

—0.  73  102 

''        wt. 
fo.8i       3 

M.85      5 
-1.29      5 
+0. 75      3 
—0. 65      4 
-0. 35     16 
—0.30    24 
-0.37     26 

Tlu5  reiiiiiikahle  feature  of  these  rcsuItH  is  tlu>  near  approach 
to  constancy  hi  the  vahies  of  the  niunbers  in  each  column, 
after  the  secular  variation  is  allowed  for,  and  the  ]arp>  magni- 
tude of  the  corrections.  The  tuost  natural  coin-Iusion  is  that 
the  reduction  from  the  limb  of  the  planet  or  the  observed 
center  of  lijyht  to  the  true  center  was  too  small  by  an  amount 
which,  at  the  mean  distance  of  the  Sun,  must  have  been  nearly 
or  quite  a  second  of  arc  {cf.  ^  3).  The  adopted  semidiameter 
3".4  seems  so  well  established,  both  by  micrometric  measures 
and  by  heliometer  measures  during  transits  of  Mercury,  that 
such  a  correction  to  the  diameter  seeuis  iuadmissible. 

1  have  not  yet  been  able  to  enter  upou  the  investigation  of 
the  source  of  this  anomaly.  A  very  im]>ortant  (piestion  is  that 
of  its  influence  on  the  results,  i^ixwe  a  constant  error  in  the 
radius  vector  of  a  planet  would  have  opposite  eflects  on  the 
elements  in  different  i)oints  of  the  relative  orbit,  it  nniy  be 
inferred  that  the  effect  of  the  error  would  be  nearly  elimiiuited 


»■■" 
V 


S^i,  m\    CUMPAUiaoN  OF  OUHKHVaTION.^  OP  MKIUTUY.  fll) 

in  III!  e.\t»MiHiv«'  series  of  obseivation.s  <li8tributi'<l  ('«nially 
between  the  two  oloiij^iitiona.  A«;tniill.v,  however,  tliere  seems 
to  liave  iieen  an  appreeiabh^  lacli  of  Hynimetiy  in  this  respeet, 
as  tlie  intluenee  of  tlie  unl<nown  qnuntity  npon  the  otlier 
unknowns  is  not  inconsitlerabhs  Altlio'ijrl'.  tlie  law  of  ehan^'e, 
as  shown  in  tlie  preeeilini;  table,  does  not  eonespond  to  the 
nia^'nltnde  of  the  eoelli<'ient  of  6r",  this  coeMlcient  being  rela- 
tively too  small  near  inferior  conjunetion  and  Unt  hwa^)  near 
snperiur  conjunction,  it  is  still  probable  that  through  the  intro 
duction  and  elimination  of  6r"  a  large  part  of  tlu^  injurious 
ettet '  is  eliminated. 

CompariHon  oftrnnxitn  and  nHridian  ohserrntionx  itf  Mercury. 

.*U.  Another  remarkable  result  which  nuiy  be  associated  with 
this  is  shown  by  the  diifereui'e  between  the  solutions  A  and  IJ, 
in  the  (taso  of  the  e(!centricity  and  perihelion  not  only  of  the 
planet,  but  of  the  Hun.  It  will  be  seen  that  the  nwridian 
observations  alone  give  a  negative  correction  to  the  ec<;en- 
tricity  of  the  planet,  while,  when  the  transits  are  included, 
the  correction  becomes  positive.  That  this  is  due  to  a  system- 
ati«^  cause  runniug  through  the  observations  is  shown  by  the 
fact  that  the  same  thing  is  true  of  the  secular  variation  of 
the  eccentricity.  This  relation  of  the  correction  to  its  secular 
variations  hohls  true  for  three  of  the  four  relative  elements, 
and  for  the  eccentricity  and  perihelion  both  of  the  planet  and 
of  the  Karth.  In  the  case  of  the  Earth's  perihelion,  however, 
there  is  a  nearer  approach  to  conformity  between  the  two 
results. 

There  is  yet  another  anomaly  in  this  connection,  which  indi- 
cates a  very  considerable  systenuitic  ernjr  in  the  ohh^r  meridian 
observations,  which  is  not  completely  eliminated  from  the  ele- 
ments. If  we  take  the  values  of  the  unknown  4|uantities  and 
their  secular  variations,  whicdi  result  from  the  two  solutions, 
ami  substitute  them  in  the  liius-ir  functions  of  the  corrections 
to  the  elements  derived  from  the  transits  alone,  namely 

V  =  1.487  61  -  0.487  dn  -  l.VM  Se  -  1.01  M"  +  \.\\)r"6n" 

-f  1.58rfr" 

W  =  0.710  61  -f  0.284  dn-  4-  0.896  6e  -  0.07  61"  -  1.11  e>'67r" 

-  1.62  6e" 


1 


I*  ■ 


70  MERCURY,   VKNI  S,  AND  MARS, 

we  find  tlu'  follow  j:>g  results: 


[34, 35, 36 


From  meridian  observations    V  =  -  2".00  +  0".fi9T 
Prom  November  transits  -  1   .(iO  -  2  .03  T 


From  combined  solution 


2  .30  T 


From  meridian  observations  W  =  -f-  0".<SJ>  —  4".."i5T 
From  May  transits  alone  +  1   .39  +  1  .84  T 

From  combined  solution  +1    .39  4-0  .42  T 

We  eoin'liide  tliat,  had  no  transits  ever  been  observed,  tl'O 
errors  of  the  eh'ments  and  their  secular  variations,  derived 
from  tlie  j-reat  mass  of  meridian  observ.atious,  would  have 
caused  an  error  of  some  ."»"  per  century  in  the  heliocentric 
place  of  the  planet  at  the  times  of  the  May  transits,  and  of 
some  3'  at  the  time  of  the  November  transits. 

The  fact  that  the  combined  soluti<m  B  satisfies  the  transits 
so  niu<'h  better  tlian  A,  althou*;h  the  total  weij;ht  of  ecjuations 
A  is  so  much  };reater  than  that  of  the  transit  e(|uations,  shows 
that  the  meridian  ttbservations  j;ive  only  weak  results  for  the 
functions  in  question. 

Mi'ridhni  ohsvr  rat  ions  of  Vphus, 

35.  So  far  as  tlie  meridian  observations  are  concerned,  those 
of  A'enus  were  treate<l  on  the  same  general  plan  as  the  observa- 
tions of  Mercury.  The  following  are  the  ]>rincipal  points  of 
dirt'erence: 

1.  The  hypothetical  (luautity  i^r'-'  is  omitt«'d.  Hence  no 
iiulex  tt)  the  consistency  of  the  observations  at  <litierent  points 
of  the  relative  orbit  can  be  derived  from  the  solution. 

2.  Tenths  of  a  unit  were  incbu'eil  in  the  coeiticients  of  the 
equations,  and  no  moditi(;ation  v,us  nuide  in  the  units.  The 
units  and  tenths  were,  however,  dropped  in  the  final  solution 
of  the  nornud  e(|uations. 

h'rsnlts  of  ohavrred  tronsits  of  YenuH. 

30.  We  put,  at  the  time  of  a  transit, 

r,  the  longitude  in  orbit  of  Veiuis; 
i,  its  mean  longitude,  or  the  mean  vame  <>f  p; 
/?,  A,  its  ecliptic  latitude  and  longitude; 
L,  the  Sun's  true  h)ngitudec 


36 1   EQUATIONS  OF  CONDITION  FKOM  TRANSITS  OF  VENUS.     71 
Then 

6\  =  cos   i  6v  +  sin'-'  /  rf^ 

•=  0.01W2  6r  +  iwrm  isiii  i  dH 

Wo  thus  have,  lor  the  dates  of  the  observed  transits, 

1701- 09 ;  6fi=-  0.0ri!)2  (Sr  +  0.9982  sin  1 6fi 
1 87-4-'b2 ;  <yi  =  +  0.0592  (S  r  _  0.9982  si  n  /  rf^ 

T  have  disenssed  very  fully  the  observations  •)f  the  transits 
of  17(>l  and  17(i9  in  AstronomUal  Paperx,  Vol.  ii.  The  tinal 
results  whieh  1  shall  use  are  found  on  i)age  404  of  that  volume. 
Jlere  I  have  put. 

.r,  correction  to  A  —  T^; 
—  I/,  eorreetion  to /^, 

the  Sun's  latitude  being  sui>|M)sed  to  require  no  corret'tion. 
The  values  of  x  ami  //  for  1709  are  clistinyuished  by  an  aceent. 
I  have  also  rei/risented  by  c..  and  ^,  the  eorreetions  to  the  dif- 
ferenee  of  the  semidianieters  of  the  Sun  and  jdanet,  tor  the 
respeetive  internal  eontnets,  to  which  may  b»'  a<hled  the  un 
known  but  ]u*obably  nearly  constant  quantity  t\\\v  to  perst)nal 
error  in  estiiMatinf«-  the  time  of  contact.  I'rom  their  very  natuie 
these  (juantitics  d(»  not  admit  of  acj'uratc  «lctcrminatioii,  and 
nuist  therefore  Im'  climinatc<l  frouj  the  i'i|uations.  Fiom  tlie 
observations  of  internal  conta<'t  are  derived  tln'  tollowiu};  four 
e([uations: 

1701     11 ;  —  .S7.r  4-  .."0  »/  -f  :,  =  -  0".07 

HI;    -f  .OS       -f  .:;{       -f.  : ,  =  —  (>".(M1 

1709     II;-  .04  .!•'  -  .77  y'  +  z^  =  -  0".27 

HI;  4  .SI     _  .r,r,    4. .-,  =  4-  o",(»2 


We  have  here  more  unknown  iiuantities  than  equations,  .so 
that  it  is  not  pratfticable  to  determine  them  all  separately. 
What  I  have  dcuie  has  been  llrst  to  assjinu'  z^  =  r,.  This  pre- 
supposes that  the  vlistance  of  centers  at  the  estimated  appa- 


\ 


72 


MEBCT  RY,  VENUS,  AND  MARS. 


[36 


rent  coiitatrt  tit  t>t;r»ss  is,  in  tlio  {j^eiuMal  incan,  the  .sanu*  as  iit 
injjress.  Tlio  result  of  any  error  in  this  liypothesis  will  be 
almost  completely  eliminated  from  the  mean  latitude  at  the 
two  transits,  but  not  from  the  longitude. 

Still,  the  values  of  .i-  and  //ean  not  be  sejiarately  deteimined; 
I  have  tiierefore  so  combined  the  e<|uations  as  to  obtain  mean 
values  of  .!•  and  y  for  the  two  ('ontacts,  assuming  that  this 
would  be  the  result  of  sui»i»osin};  these  i|uantities  to  havt^  the 
same  viilues  at  both  epochs.  Calling;  these  values  .v"  and  y", 
we  have  by  addition  an<l  subtraction,  supposing  Zj  =  ^3, 

-  {).:W.r"  +  2.55  J/"  =  0".11» 
.'UW.r"  +  0.45  V"  =0".30 


ii 


We  thus  have* 

If"  =  -f  ()".(M) 

These  corrections  are  not  api)licnblc  to  the  coordinates  from 
Lkvekrikr's  tiiblcs  as  they  stan*!,  but  to  those  ipiantities 
as  corrected  by  the  following;  amounts: 

J\z=  +  0".L'5 


"  III  R  Hecouil  n|iproxiiiiiiti()ii  to  thust-  (|iiaiititi<'H,  whioli  may  !)<>  luailc 
nftiT  thii  corn'ctioii  to  tlie  nMitonnial  iiiotion  nf  tbo  mule  in  dcternuiu'd, 
wo  HhuiiUl  put,  oil  account  of  thin  oorrectiou, 

,/  =-_,/'  -O'.ll 
,/^y    +0".U 

The  Holution  svoiild  tlu-u  give 

y"  =  +  0'.Ofi 

X"— .  +  0  .14 

I  huv«)  carriiMl  through  a  iiioru  *  laeful  iip|iroxiinatioii  in  a  aii)i8oi|Uont 
obuptf  r. 


[36 


30]  El^VATlONS  OF  CONDITION  FROM  TRANSITS  OF  VKNUS.    73 

We  tlins  HihI,  for  tbc  coiToctions  to  Levekkier's  tables  at 
the  e|M)ch  l?l}.").5, 

(5  A  —  rfL  ==.  +  0".01>  4-  0".2r>  =  4-  «".34 
6fi  =  -  0".(M>  +  2".(H)  =  +  1".»4 

and  lieiicc 

Bin  J  fJ  ^  =  4- 1".!).")  +  0".O.VJ  rf  L 

A  stir  farther  iiioditicatioii  is  riMiuired  to  the  taludar  loiif;!- 
ttule  on  uccoiiiit  of  the  correction  to  tlie  mass  of  the  Harth 
used  by  LEVERitiKil,  a:id  hence  U^  the  periodic  perturbations 
in  ion{;itude.  This  correction  is  -f  (>".2().  Wo  thus  have  for 
tlie  correction  to  the  orbit  W>ngitu«h'  of  V«'nuH— 

rfr  =  +  0".02  +  0".!)<»8  r?  I, 


For  the  results  of  the  transits  of  1S7J  and  1882  f  have 
de]>ended  entirely  on  the  helioineter  measures  and  photo- 
jfrajihs  njade  by  the  (ierir\an  and  American  expeditions, 
respectiv«'ly.  The  «lellnitive  results  of  thetrernnin  observa- 
tions, as  \vork«'d  up  by  Dr.  AuwEKs,  are  found  in  \'ol.  V  of 
the  (xerman  Heports  on  the  Transits.*  The  American  ]ihoto- 
j^ra',)hic  measures  of  1S74  have  not  been  otiicially  worked  up 
and  published,  but  a  preliminary  investi};ation  from  the  data 
contained  in  the  pubhshed  nieasures  was  male  by  1).  I'.  Todd, 
and  publisiied  in  the  Amrrivan  'lournnl  of  Sru-nw,  Vol.  21, 
18S1,  pa«;e  4!M.  Tiie  measures  of  1.S.S2  have  b<cn  dellnilively 
work«Ml  up  l)y  IIaiiknkss,  l>ut  only  the  results  published. 
They  are  tbund  in  tiu^  report  of  the  Superintendent  of  the 
U.  8.  Naval  Observatory  for  the  year  IHUO. 

Tile  corrections  to  the  ;i('ocentric  Kijrht  Ascension  and 
l>eclinati<Mi  of  Venus  relative  to  the  Sun  tlius  derived  are 

•  Dl«  V<Mins-iliirrhK.in>{t)  1S7I  innl  IHSi*  Ituricia  ubiT  «lio  iX-utmheu 
]<0(»l»iicliliiaK<)ii  KiUit'tor  Mnn«l,  Hurliu,  181i:{. 


74 


MERCURY,  VENUS.  AND  MARS. 


given  ill  the  following  table.  In  taking  the  mean  the  weights 
are  not  strictly  those  which  wouhl  result  from  the  probable 
errors  as  assigned,  but,  in  accordance  with  a  general  princi- 
l)le,  independent  results  have  received  a  weight  more  near  to 
e(iuality  than  would  be  indicated  by  the  mean  errors. 


1874:  German,         d  K.  A.  =  +  4.77  t  0.28 
American,       .      .      .     +  4.14  i  0.30 

Adopted,    ...-}-  4.44 


pi      " 


I      ; 


German,        6  Dec.  =  +  2.28  i  0.10 
American,      .      .  -f  2.50  i  0.30 

Adopted,    .      .      .     +  2.34 

1882:  (German,        rf  R.  A.  =  +  0.03  1  0.12 
American,      .      .      .     +0.101-0.08 

Adopted,    .      .      .     +0.07 


German,         d  Dec.  =  + 2.02  ±  0.00 
American,      .      .      .     +  2.02  ±  O.OS 

Atlopted,    .      .      .     +2.02 

We  change  these  results  successively  to  geocentric  longi* 
tude  and  latitude,  heliocentric  longitude  and  latitude,  and 
orbital  hmgitude  and  latitude.  The  results  of  these  several 
changes  are  as  follow: 


1874 

1882. 

Corr.  in  gcoc.  h)ng. 

+  3''.a53 

+  8".077 

Corr.  in  hit' 

+  2  .724 

+  2.  071 

Corr.  in  hel.  long. 

-1  .415 

-2  .965 

Qorr.  in  hcl.  lat. 

+  I  .(M)l 

+  1  .001 

Corr.  in  orbital  long. 

-  I  .35 

-2  .90 

Value  of  sin  i6  6 

-I  .26 

t   i 


37 J  EQUATIONS  FROM  TRANSITS  OF  VENUS.  75 

Eqwit  ions  from  transits  of  Venus. 

37.  The  corrections  to  the  heliocentric  ])ositioii8  of  Venus 
and  the  Earth,  as  thns  found,  are  now  to  be  expressed  in 
terms  of  corrections  to  the  elements.  The  results  of  this 
expression  are  shown  iu  the  following  equations: 

Equations  f/iren  by  the  corrvctions  to  the  orbital  longitude, 

I.  Epovh,  17(m.5;  t-  -3.90;  weight  =  20() 

0.902  (H+IM  e^TT  +  1.62  rfe  -  0.970  61"-  1.81  ('"d?r"-  0.85  6e" 

=  +0".02  },  0."15 

II.  Epoch,  1874.9;  r  =  +  0.48;  weight  =  400 

-  0".88//  +  \.mt  (U  -   1.223  <'rtV  -   1.590  6e  -  l.(>30  61" 
+  1.8(»4  e"6n"  +  0.817  6e"  =.-.  —  1".35  i  0".08 

III.  Epoch,  lS82.th  r  =  +  0.80;  weight  =  8(M) 

0".<;0//+ 1.008 61  -  1.140  e6n  -  1.051  6e  -  1.028  61" -\- 1.825  e"6n" 

-f  0.900  6e"  =:  -  2".!M)  i  0."027 

Equations  (jircn  hif  the  corrections  to  the  orbital  latitude. 

1.  1765.5;  sin  i6fi- 0.057  61" -O.ll  e"d;r"— 0.05 6e"=z  +  1".95 

i  O'.IO 

II.  1874.!>;  sin^J^-0.06l()7"-H0.110e"fJn'"+0.018f>\"=  -  1".08 

I   0".04 

Ml.   1882.9;  sin/rfW-0.061  fy/"-}-0.l07e"rf;r"-f  0.()53rt\'"  =  -l  ".26 

i  0".0l9 

The  weiglits  assigned  to  these  three  equations  are,  respec- 
tively, 200,  600,  iiiid  l,(i00. 

Before  using  these  ecinations  tlie  corrections  to  tiie  elements 
were  transformed  into  Hie  unknown  (|naiitities  dilincil  in  i'i', 
and  their  secular  variations  by  multiplying  the  coeOicientf  by 
the  factors  given  on  page  56. 


70 


MEKCl'UY,   VENl  H,  AND  MAKS. 


138,  39 


SolulioHH  of  the  equationM  for  Venus. 

38.  The  ptirtH  of  tliu  uoriiial  eiiiiutiuiiH  furiiUHl  fVoin  the 
pi-ecediii^:  conditional  equations  were  added  t4)  tlie  parts  from 
tlie  meridian  obHervations,  and  tlie  resulting  solution  H 
obtainid.  Ah  in  the  case  of  Mercury,  a  solution  A  was  made 
of  the  nornuil  ei|uations  derived  from  the  meridian  observa- 
tions alone.    The  results  are  as  follows: 

VKMS, 
HchuUh  of  MolutiouH  It/  the  normal  equatUnktt. 


Unknowns. 


Svml)ol. 


/ 

e 

f 

t" 

n 

I 

.1 
N 


t 


15. 


u. 


-0.0834      7. 


— o.  0708 

I  -o.  143s  -o.  1501       5. 

i  +0.  1156  40.  IJ40       6. 

j    jo.  0164  f  0.0106  j     7. 

1    f  o.  0941  10.  1003  I     3. 

jo.  0628  t  o.  0764  I     3. 

f  0.0246  |-o.  0271  I     4. 

fo.0336  (-0,0318!     2.5 

0.0274         ~0. 02I2    \      2. 

40.4742  1  0.4642    i       I. 

-o.  03S3  -o.  037s       5. 

—0.0768  -o.  0743       4. 

— o.  1846  — o.  1983  20. 

4-0.0970  jo.  1088  :  34, 

-0.0561  i      0.0594  28. 

4-0. 1472  !  4-0, 1644  12. 

1 0.0555  10.0698  12. 

f  0.0182  4-0.0202  16. 

i  0.0283  4-0.0317  10. 

.  ().  0399       i  o.  0506  I     8. 

-n.  0820         o.  0347       4. 

0.0020         0.0002  2o. 

-0.056a        -o.  o6(»2  16. 


CniT«ction.<i  uf  dements. 

Symlxil. 

A. 

It. 

t\  m :  m 

-0.  496 

-0.  584 

ff 

/' 

A I 

-0.718 

-0.751 

''J 

4-0.   r.94 

•  0.  804 

sin  I. IN 

4  0. 115 

i  0.  074 

+0.  282 

4-0.  301 

-•<»r 

4-0.  188 

f  0.  229 

At 

4-0.098 

4-0.  108 

f>t" 

4-0.084 

40.  080 

."  &  If" 

-0. 055 

-0.  042 

1 

-f-o.  474 

-i  0.  464 

,\ 

—0. 192 

-0.  18S 

M" 

—0.  307 

-0.  297 

I ),  r'  / 

—3.  692 

-3-  y66 

I'.  1 

4  2.  328 

f  2.  Oil 

sin  1  1 ),  N 

-  '57" 

I.r.63 

i>,. 

-f  1.766 

f  '••>7J 

M),r 

-1-0.  666 

40.838 

1).« 

4-0.  291 

+0.  323 

I),.-" 

4-0.  2S3 

(0.317 

r"lJ,T" 

-f  0.  3«9 

-f-o.  405 

I  >,  (1 

~o.  328 

0.  139 

I),  a 

-0.  040 

-  0.  004 

1 ),.»/" 

—  0.  899 

-1.059 

Mean  epoch  nf  correction,  1863.0 

CouiixfriHou  0/  tniHHitu  0/  Vt-nHH  icith  meridian  ithHtrrationx, 

:v,)  To  .show  to  what  extent  Ihf  resnUs  of  the  meridian 
observations  ditfei  iVom  tho.stMtt' th««  observetl  transits  over 
the  Sun,  we  {\\\u\  tlie  xidnes  of  the  tibsolnte  terms  of  the 
e«|iiationM  of  condition,  §.'17,  tlrst  by  substituting  the  values 
A  of  the  corrections,  and  tht>n  the  vallu^s  H.     We  thus  have: 


I 


J  1 


39,  4(>|  K«,HATroNS    KUOM   TUANSITS   Oi"    VHNUS. 

lieHulualH  in  orbital  huijitude. 


(a)  From  iiu'iidiaii  ol»s.  aloiu' 

( (i)  From  comhiiu'd  Holiitioii 

{y)  From  transits  alone  .     . 

Discordance,  (;')  —  (")    • 

Discorilancu,  (v)  —  (li)    . 


«76S  5 

-  0  ".07 

-I-  WM 

4-  (Y'SYl 

-|-(>".01> 

—  ()".(>» 


lS74'i. 

-  V'M 

-  \".\:\ 

-  1  '..'{o 
-f  0  ".01 
4-     O.OH 


UixitinalH  ill  orbital  latitihli: 


{it)  I'rom  meridian  ohs.  alone 
(fi)  From  coiid>lned  solntioii 
(r)  From  transits  alone  .     . 

hisi'oidance,  (r)  — (<>)    . 

Discordance,  ()'j  —  (fi)    . 


'7"5-5- 
-I-  I '".02 

+•  J'.OU 

4-  i".o:» 
4-  o'".o:{ 
-(►".11 


i.S74.'». 

-0"'.77 
-0'".!U 

—  l".o,s 

-  o"'.;jl 
-0".17 


77 


lSS2.n. 

-  Ii"'.."i4 

-  L'".7H 

-  '2" AH) 

-  {)":M 
-0'".I2 


1S83.9. 

-  o".m} 

-  l"".!:-' 

-  I  ".'-m; 

-  o"".;{(» 

-0".14 


It  will  b«'  seen  that  the  comhined  solution  represents  the 
ohservatioMs  of  th«>  transits  mucli  lietter  here  than  in  the  cuku 
of  .Mer«niy. 

Solution  0/ ihr  niuations  for  Mars. 

40.  As  the  formation  of  the  normal  eqnations  for  Mars  was 
a|i|)roachinu  its  end,  a  singular  discordance  anion;;  the  resid- 
uals of  tlic  paiiiiil  normal  ci|uations  for  ditVerent  perioiKs  was 
noticed.  On  traeinp;  the  matter  out  it  appeared  that  while  the 
correction  of  riie  «eocentri«^  lon^fitude  of  liKVHititlKK's  tahles 
in  1H4.'>  and  a^ain  in  I.S'.rj  was  i|uit«'  small,  the  correction  in 
ISO12  wan  consideral>le.  Now  theie  i.s  an  iiio<|Uality  of  lon^' 
l>eri«»d.  about  forty  years,  in  the  mean  m«)tion  of  Mars,  depend - 
iuffonthe  action  of  the  lOiirth,  and  havin;::  for  its  ar;;innent 
ITu/'  — 8f/.  This  coeni4ient  isof  the  st^venth  (U'dcr  in  the  eccen- 
tricities, and  the  terms  of  tin'  ninth  or  even  of  the  eh'venth 
order  ndffht  he  sensible  in  a  dcvclopnu-nt  in  powers  of  the 
eecentriciti«'s  and  sines  or  cosines  of  nndtiples  of  the  mean 
louj^'itudes.  The  conclusion  which  I  reached  was  that  the  the- 
oretical \alue  <d'  l\i'\>  coelliricnl  u as  not  determined  with  suill- 
dent  precision.  As  the  wtak  of  solvint;  the  e(|uations  could 
not  wait  for  a  lu'w  determinati(m  and  a  im*w  formation  of  the 
absolute  terms  of  the  normal  c(|uati<nis,  it  was  decided  to  make 
an  approximate  empirical  correction  to  the  thc<u'y.  This  was 
used  to  coiled  the  aiisoliite  ti'rins  of  the  partial  normal  etjua- 


MEUCIBV,   VENUH,   AND   MARS. 


(40 


n 


tioiis  for  (>iu'h  perJoil,  uiid  the  solution  wuh  thou  proceeded 
with.  The  (;hiiiiceA  se<'in  to  l>e  that  by  this  prm^ess  the  iiijii- 
rioiiH  el!e(!t  of  tlie  error  up(»ii  the  ehMuents  derived  from  the 
equations  would  he  inconsideruble;  thiH  is,  however,  a  point 
on  whieh  it  is  impossible  to  speak  witli  certainty.  It  is  the 
intention  of  tiie  \vrit4>r  to  recompute  the  doubtful  terms  of  the 
peiturbations,  and,  if  ixissible,  reconstnu;t  the  absolute  terms 
of  the  niuiual  eipiations  in  accordance  with  the  c(U'rected 
theory.  Meanwhile,  the  present  work  nei-essarily  rests  on  the 
imperfect  theory  with  the  approximate  empirical  corrections, 
which  are  as  follow: 

61  =  U":60  cos  (15f/'  -  Hfi  _  223^) 
Mtt  =  O'Mo  cos  (1%'  -  Hji) 

As  the  elements  of  Mars  are  derived  wholly  from  nu'ridian 
observatifuis,  only  one  set  of  cipiations  of  condition  was  formed. 
The  results  of  the  solution  are  shown  in  the  following  table: 

MAKS. 


rnkiiowns. 

^ 

f'i>rrcction.s 

nf  c]cment>. 

Symhol. 

N'iilue. 
—.02278 

03 

Symbol, 
ft  •« :  m 

Value. 

['"'1 

1 

-0.  0  17 
// 

/  ] 

-.44S54 

2. 

.5/ 

0.897 

;•'  ■ 

-f.  05479 

2-5 

'M 

+0.  137 

,  ^ '. 

f-.of>724 

2-5 

Sin  1. IN 

+0.  168 

'  <■ ' 

+  •4380.? 

1  II 

r 

fo.  626 

T 

— ,0505() 

U  ij 

,\z 

—0.  722 

f 

f. 07474 

4. 

iSr 

+0.  299 

1 

-. 4989S 

2, 

i\e" 

-0.  998 

'_'/' 

- . 42409 

2. 

<-"<)t" 

-0.  848 

(t 

+  .1S545 

5- 

a 

-f  0.  927 

■    l\    ' 

— •  045.?<> 

5- 

i\ 

—0.  227 

'-"'' 

+  .05786 

i- 

,\l" 

+0.  174 

■     /    ■ 

-\-.  16605 

8. 

I),<1/ 

+  1.328 

;  J 

t-.  1340S 

10. 

1».J 

4-I-34I 

;  N ; 

—.02265 

10. 

SinJD.N 

-0.  226 

(* 

-.03180 

S« 

l\c 

—0.  182 

7T 

— .  00928 

lyu 

D.r 

-0.  530 

t 

1  .  o6o()7 

16. 

\hr 

+0.  976 

c" 

-•  "2597 

s. 

\h.-" 

—  1.008 

'_//' 

+  .0*^853 

8. 

r'\\j" 

fo.  068 

(1 

—.09670 

20. 

l),a 

-1-934 

.) 

—.01168 

20. 

I),.l 

-0.  234 

/"; 

'  . 13111 

1 2 

D,.!/" 

+  >-S73 

140 


•    411 


REFERENCE  TO  THE  ECLIPTIC. 


(9 


HefervHce  to  the  ecliptic. 

41.  In  all  the  precedinj;  (leterriiiiiatioiiH  the  planes  of  the 
orbits  are  referred  to  the  plane  of  the  ICartli's  i>(|iiator,  or,  to 
speak  more  exactly,  to  a  phine  through  the  Snn  parallel  to  tin* 
lOarth'H  e(|uator.  Ah  in  aHtrononiical  practice  the  ecliptic  is 
tai^en  as  the  fiiiulamcntal  plane,  it  is  necessary  to  investiKute 
the  reduution  of  the  elements  from  one  plane  to  the  other. 

Let  us  consider  the  spherical  triant;le  formed  on  the  celestial 
sphere  by  the  plane  of  the  orbit,  the  pl.me  of  the  ecliptic,  and 
the  plane  of  the  Karth's  equator.  For  the  sides  and  opposite 
angles  of  this  triangle  we  have 

Sides:  }S(  B  if 

Opjwjsite  angles:  /  1S(P  —  J  f 

When  equatorial  coordinates  are  used,  the   position  of  the 
planet  is  considered  as  a  function  of  the  three  (quantities 


N|       Jj 


(") 


When  ecliptic  coordinates  are  used,  the  three  corresponding 
quantities  are 


e^      ii 


(i) 


Taking  the  set  of  quantities  (a)  as  the  fundamental  parts  of 
the  triangle,  and  expressing  the  »'orre«!tions  of  the  otiier  parts 
as  functions  of  them,  we  have 


Si=  ■{■  cos //'«yj  +  sin  »/•  sin  Jrf X  —  cos  Hfif 
sin  iS6  =s  —  sin  i/^Sf)  4-  cos  »/'  sin  JrfN  -{-  cos  /  sin  Wdf 


(c) 


Taking  (h)  as  the  fundamental  parts,  we  have  for  the  correc 
tions  to  N  and  J 


sin 


ff  J  =  cos  yrfj  —  sin  '/'  sin  /rtV/  -f  cos  Ndf 
.IrfX=  sin  i/'6i  -\-  cos  //•  sin  iSH  —  cos  .1  sin  NfJf 


«/) 


The  numerical  values  assigned  to  the  eoeflleients  in  these 
eqinitions  are  those  corresponding  to  the  mean  epoch  ISM). 
The  fact  that  they  change  somewhat  in  the  course  of  a  hundred 
years  has  not  been  taken  account  of.  The  future  astrontuner 
will  meet  with  a  real  dilliculty  iu  that  the  corrections  to  a 


Hi) 


MKItCUBY,   VENl'M,   AND   MAU8. 


[41 


ii 


svt  of  cUMiiciits  at  Olio  eiH)('.h  «Io  not  iiccumtoly  corrcHpoiul  to 
Hitniliir  (MirrtM'tioiiH  at  anotlier  e|M>c)i.  It  is  iinpossiblo  to  do 
away  ligoioiiHly  with  tin'  liitliciilty  thus  arisiiij;,  oxoept  by 
iiitrodiu'iiii;  a  iiioro  ^ciicral  HyHttMii  of  oleinuiitH  than  elliptic 
onoH.  Tlu;  error  is,  happily,  not  important  in  the  present  statu 
of  astronomy.  The  equations  in  question  for  the  three  planets 
are  as  follow : 

,U  =  4-  .71«>  fU  -I-  .«»0L*  sin  .1  rJ  N  —  A\HH  fSf 
sin  jrtvy  =  -  .OOU rf.I  -I-  .71M»  sin  J  rf  N  4-  .7L'l  6e 

VenuH. 

fj;  =  +  :m:\  (^  .i  +  .tn's  sin  .i  ti  X  -  .l*.m  rf* 

sin  i6&  =  -  .1»28  fif  .1  4-  .•{T.'t  sin  .1  f)'  N  +  .JM»7  fJ* 

6i  =       .70.'{  rf  J  +  .IVJ  sill  .1  rV  N  -  .«i<U  fJ* 
sin  i(W  -  -  .711i  fJ  J  +  .7().{  sin  J  rf  N  +  .747  de 

For  the  inverse  relations  we  have — 

Mtrrurif, 

(?.!=:  .IW)  fU  -  .(MH*  sin  iSH  +  .*Mi  »Se 
sin  .1  rf  N  =  .<;02  fU  -f  .7!)U  sin  »"rJ#  -  .H»L»  rff 

r)M  =  .;J7.'J  rf/  -  .IH'8  sin  i6H  +  .«MM)  rff 
sin  J  fJ  N  =  .!L'.S  6i  +  .373  sin  166  -  .125  rff 

(U  =  .70;{  fj/  -  .711.'  sin  i>SH  +  .1M»8  rff 
siu  .1  <S  N  =  .711'  6i  +  .70;}  sin  itUi  -  .o:)2  df 


CIIAITKU  IV. 

COMBINATION  OF  THE  PRECEDING  RESULTS  TO  OBTAIN 
THE  MOST  PROBABLE  VALUES  OF  THE  ELEMENTS 
AND  OF  THEIR  SECULAR  VARIATIONS  FROM  OBSER- 
VATIONS  ALONE. 

In  tilt'  two  pn-nMlitifr  rluiptors  an>  dorivi'd  four  Hoparate 
valiu's  of  tlu^  six  roin'ftioiis,  <r,  rt\  rtV,  rt7",  or",  and  v"6n'\  and 
of  tlu'ii-  s«><'iilar  variations,  which  pertain  to  the  orbit  and 
motion  of  the  ICarth  rehitivo  to  th<'  stars.  \V«'  hav«'  now  to 
couibinc  tlu'so  fonr  resnits  so  as  to  diMive  tlic  most  probable 
valnos  of  thf)  twi'lvo  unknown  ijuantities  in  ipifstion. 

IhriotionH/rom  the  imthod  of  least  Hijiuircti. 

42.  If  W(>  applii'd  witliont  inodillcation  tht'  principles  of  thu 
mi't hod  of  least  s<|uares.  we  sliould  first  eliminate  the  elements 
and  secular  variations  for  <'a(;h  planet  from  the  normal  equa- 
tions {iiveii  by  observations  of  that  plaiu't,  which  would  leave 
us  with  three  sets  of  nornml  equations,  eontaininf;  only  the 
twelve  quantities  depending  on  tln^  motion  of  the  Karth.  We 
should  then  reduce  these  uornuil  eqtnitinns  to  equality  of 
weight,  by  multi|)lying  each  of  them  by  the  appropriate 
factors,  and  we  should  then  consider  the  observi'd  corrections 
to  the  solar  elements  <U^rived  from  observatituis  of  the  Huu 
alone  as  affording  eijuations  of  condition  to  be  reduced  to  the 
adopted  system  of  weights,  and  then  midtiplied  by  their  coetli- 
cients  and  added  to  the  normal  equations.  The  solution  of 
the  single  set  of  normal  equations  thus  formed  would  lead  to 
the  definitive  values  of  the  solar  elements  and  of  their  secular 
variations,  which,  being  substituted  iu  the  eliminating  equa- 
tions from  each  ])lanet,  would  lead  to  the  detiuitivo  elements 
of  the  planet  and  of  their  secular  variatious. 

This  proceeding  is  not,  however,  advisable  in  the  present 
case,  bee-iiise,  owing  to  the  immense  mass  of  material  worked 

um  ^   ALM 6 

81 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


/q 

^  ^t^ 


1.0 


I.I 


l^|2£    |2.5 
1^  1^    |2.2 

£:   b£    12.0 


1.8 


1.25  ||U  ,,.6 

< 6"     

► 

Photographic 

Sdences 

Corporation 


23  WfST  MMN  STREET 

VV.^BSTBIt.N.Y.  MSSO 

(7^4)  8724503 


82 


puohability  of  errors. 


[42 


¥i  i- 


Uj),  the  errors  to  be  principally  feared  are  not  the  accidental 
ones,  of  which  alone  the  method  of  least  squares  takes  a<!Coimt, 
but  the  systematic  ones  arisinj^  principally  from  personal 
equation  and  imperfect  reduction  of  the  observations  to  the 
actual  center  of  the  planet  or  of  the  Sun.  These  errors  aifect 
different  elements  in  very  different  ways  and  to  different 
amounts;  from  some  they  will  be  almost  completely  elim- 
inated and  from  others  they  will  not.  We  must  tlierefore  pro- 
ceed by  a  tentative  process,  ascertaining  at  each  step,  so  far 
a^  possible,  how  each  result  will  come  out  before  we  accept  it 
as  final,  to  be  (combined  with  other  results.  In  doing  this  it 
is  necessary  to  deviate  so  widely  from  what  are  conuuonly 
regarded  as  fundamental  principles  of  the  theory  of  the  com- 
bination of  observations  that  a  brief  presentation  of  the  prin- 
ciples involved  is  appropriate. 

It  is  frequently  accepted  as  an  axiom  that  when  we  have 
several  non  accordant  determinations  of  the  same  (luantity, 
between  which  we  have  no  reason  for  choosing,  the  most  prob- 
able value  is  the  arithmetical  mean.  The  operation  of  taking 
the  arithmetical  mean  is,  in  fact,  the  simplest  application  of 
the  method  of  least  squares.  The  fundamental  hypothesis  on 
which  this  method  rests  is  that  the  probability  of  an  error  of 
magnitude  ±  j?  is  given  by  the  well-known  exponential  equa- 
tion 


(p  {h,  x)  dx  = 


y/  n 


dx 


(a) 


•i 


h,  the  modulus  of  precision,  being  a  constant.  It  was  shown  by 
GAUS8  thfit  this  function  for  the  probability  follows  rigorously 
from  the  principle  of  the  arithmetical  mean.  It  therefore  fol- 
lows that  the  method  of  the  arithmetical  mean,  and  therefore 
that  of  least  squares,  is  rigorously  correct  only  so  far  as  the 
law  of  error  is  expressed  by  the  above  exponential  function. 

It  scarcely  needs  to  be  pointed  out  that,  as  a  matter  of  fact, 
the  law  of  error  in  question  is  not  true.  Not  only  so,  but  in 
astronomical  experience  it  deviates  from  the  truth  in  a  way 
admitting  of  precise  statement.  It  presupposes  that  the  mod- 
ulus of  precision  is  a  determinate  quantity.  Were  this  the 
case,  then,  to  take  a  single  instance,  the  probability  of  an 


142 


421 


PHOHABLE  ERHOUS   AND  WEIGHTS. 


83 


error  five  times  as  great  as  the  probable  error  would  be  less 
than  0.001,  and  the  probability  of  an  error  six  times  as  great 
would  be  about  0.0001.  This  is  not  true,  beeause,  takiug  the 
function  (p  {h,  x)  as  a  basis,  we  may  say  that  the  modulus  of 
precision,  /t,  is  nearly  always  in  practice  an  uncertain  (juau- 
tity.     Let  us  then  put 

«!,      h>j      «;),  .  •  • 

for  the  possible  values  of  h,  and 


(«) 


Pi,   Pi,   P3,       .       -       . 

for  the  several  probabilities  that  //  has  these  respective  values. 
Then  the  probability  function  will  become 


<P  {^)  =Pi  (p  {K  •^•)  +  Pi  ^  i^h,  .'■)  4- 


(&) 


Now  this  form  can  not  be  reduced  to  the  form  (a)  with  any 
value  whatever  of  the  modulus  h.  If  we  make  the  closest  rop- 
reseiitation  possible,  we  shall  have  a  curve  in  which  small 
values  and  large  values  of  x  are  relatively  less  probable  as 
compared  with  the  facts  than  are  intermediate  values.  To 
show  that  this  is  the  actual  case,  let  us  suppose  that  we  have 
three  determinations  of  an  unknown  quantity.  If  we  pro(;eed 
in  the  usual  way,  we  should  infer  the  value  of  /<,  the  measuie 
of  precision,  from  the  discordance  of  these  three  values.  But 
it  is  evident  that  this  determination  of /t  would  be  very  uncer- 
tain. Should  the  three  values  chance  to  be  fortunately  acccnd 
ant,  then,  proceeding  in  the  usual  way,  our  function  would  lead 
to  the  conclusion  that  the  juobabil'ty  of  an  error  of  a  certain 
nmgnitude  in  the  mean  was  very  small,  when,  as  a  matter  of 
fact,  it  might  be  very  considerabh'.*     The  value  of  /(  being 


*  To  take  a  simple  ami  quitw  possihle  iiiHtance,  let  three  observations  of 
a  star  with  a  meriiliau  circle  j^ive,  for  tlie  seconds  of  ilecliiiation,  0  ".4, 0".5, 
and  0".6.     By  the  canons  of  least  8(|uares  the  mean  result  would  be 

0".50  ±  0".039 

aud  the  probability  of  au  error  as  great  as  0".l  would  come  out  about  0.08. 


84 


PROBABLE  EllRORS  AND   WEIGHTS. 


[42 


\ 


K    '• 


uncertain,  the  true  form  of  the  function  is  not  (a)  but  {b).  It 
follows  that  we  may  lay  down  the  following  general  rule: 

Tlir  brat  value  from  n  Hystem  of  non- accordant  (letermbiationa 
is  not  the  arithmetical  mean,  but  a  mean  in  which  less  weight  is 
assif/ued  to  those  results  which  deviate  most  widely  from  the 
mean  of  the  others. 

I  have  considered  the  subject  from  this  point  of  view  in  the 
American  Journal  of  Mathematics,  Vol.  VIII,  p.  ;{43,  and  given 
tables  for  determining  the  weights  to  V)e  assignecl  to  the  results 
when  the  law  of  error  is  that  derived  from  several  hundred 
observed  contacts  of  the  limb  of  Mercury  with  that  of  the  iSun 
during  transits  of  the  i)lanet. 

Another  well-known  defect  in  the  method  of  least  squares 
is  that  it  does  not  take  Jiny  account  of  systematic  errors.  The 
greater  the  number  of  observations  that  are  combined,  the 
larger  the  proportion  in  which  the  errors  of  the  results  may 
be  due  to  the  systematic  errors  in  the  observations  or  the 
elements  of  reduction.  Although  such  errors  may  elude  inves- 
tigation so  far  as  their  determination  and  elimination  is  con- 
cerned, we  may  yet  be  able  to  point  out  their  origin,  and  to 
show  to  what  extent  they  would  influence  each  separate  result. 
Of  some  results  we  can  say  with  entire  confidence  that  they 
are  but  slightly  affected  with  systematic  error;  of  others,  that 
they  may  be  very  largely  so  affected.  In  the  latter  case,  the 
weights  of  the  results,  as  determined  from  the  solution  of  the 
normal  equations,  give  no  clue  whatever  to  the  probable  mag- 
nitude of  the  error. 

The  result  of  this  is  that  in  the  following  paper  we  are  more 
than  once  confronted  with  the  following  problem:  Among 
several  determinations  of  a  quantity  one  is  known  to  be  free 
from  systematic  error  and  to  be  affected  with  a  well  determined 
probable  mean  error,  rt  f.  There  are  also  one  or  more  other 
determinations  of  which  the  probable  error  is  unknown  and 
can  not  be  determined,  because  we  have  no  sufficient  knowl- 
edge of  the  probable  effect  of  systematic  errors  upon  the  result. 
What  shall  be  the  relative  weight  assigned  to  two  such  results 
in  order  to  obtain  the  mean?  The  decision  of  this  question  is 
necessarily  a  matter  of  judgment,  the  grounds  for  which  it 
might  be  extremely  prolix  to  state  at  length.    An  attempt  has 


421 


PROBABLE  ERRORS  AND  WEIGHTS. 


85 


m 


been  made  in  these  cases  to  classify  the  results,  so  as  to  give 
a  general  idea  of  what  is  likely  to  be  their  modulus  of  i)re- 
cisiou.  and  weight  them  accordingly. 

Any  attempt  at  numerical  accuracy  in  such  an  estimate 
would  be  labor  thrown  away.  It  has  therefore  been  considered 
sutlicient  in  such  cases  to  state  what  tiie  conclusion  of  the 
author  is,  leaving  its  revision  and  criticism  to  the  future 
investigator.  Indeed,  in  some  cases,  as  in  that  of  the  correc- 
ti(»ii  to  the  centennial  motion  of  the  Sun  in  longitude,  a  con- 
venient round  rumber  has  been  chosen,  very  near  to  the  result 
of  w-ell  derermined  weight. 

We  should  be  carrying  the  preceding  conclusions  too  far  if 
they  led  us  to  a  general  distrust  of  the  conclusions  reached  bj' 
the  method  of  least  squares.  Tlu»  doctrines  that  there  is  a 
necessary  limit  to  the  accuracy  with  wliich  astronomical  deter- 
minations can  be  made;  that  systematic  errors  necessarily 
att'ect  every  such  determination;  and  that  the  canons  of  least 
S(|uares  necessarily  lead  to  illusory  ])robable  errors,  are  too 
sweeping.  Ws  .nay  lay  down  the  general  rule  that  if  we  iiave 
a  sufficient  number  of  really  independent  determinations  of  an 
unknown  (piantity,  of  which  we  individually  know  nothing 
except  that  they  are  the  results  of  actual  measures,  and  not 
mere  guesses,  then  the  arithmetical  mean  will  be  a  definite 
result,  the  probable  deviation  from  which  will  actually  follow 
the  law  given  by  the  canons  in  (piestion  with  a  closeness 
which  will  continually  iucreafo  with  the  number  of  independent 
determinations. 

If  we  have  such  knowledge  of  the  relative  vjilues  of  the 
vai'ious  detern  uiations  as  to  assign  greater  weight  to  some 
than  to  others,  the  result  will  be  s  lil  better  when  those 
weights  are  used,  provided  always  that  they  are  assigned 
without  undue  bias  in  favor  of  those  re  •ull^  which  most  nearly 
approach  the  value  suj-nosed  to  be  ai)proximately  correct. 

These  considerations  lead  me  to  a  policy  which  I  have 
always  adopted  when  it  was  easy  to  do  so  in  the  following 
discussions,  namely,  that  of  so  conducting  the  work  as  to 
lead  to  as  many  independent  determinations  of  a  quantity 
as  possible,  and  of  always  giving  a  less  relative  weight  to  such 
sets  of  determinations  as  might  from  any  cause  whatever  be 


8(5 


ELEMENTS  OP  EAUTll's   OllUlT. 


[42,43 


1 

i  1 


I:  \    I 


supposed  iirtet'ted  by  siii  important  coiuiiiou  source  of  error. 
Where  the  iii<lepeiuleut(leteriuiuations  are  few  in  uuaibcr,  the 
computation  of  a  definite  probal)le  error  is  impracticable,  aud 
the  i)robable  mean  error  assi{>iu'd  is  necessarily  a  result  of  a 
judgment  based  on  all  the  circumstances. 

Relative  pn'vision  of  the  tiro  methods  of  iletermininii  the  elements 

of  the  EartWH  orbit. 

43.  When  the  system  of  determining  the  solar  elements  from 
observations  of  the  planets  as  well  as  of  the  Sun  was  originally 
decided  upon,  it  was  supposed  that  the  two  methods  would 
give  results  not  greatly  differing  in  accuracy  tu  the  case  of  any 
of  the  elements.  This,  however,  is  i)roved  by  the  results  not  to 
be  the  case.  Attentum  has  already  been  (lalled  to  the  extreme 
consistency  of  the  values  iound  for  the  correction  to  the  eccen- 
tricity and  perdielion  of  the  Earth's  orbit  from  observations  of 
the  Sun.  This  consistency  insi)ires  us  with  conlidence  tliat 
the  probable  errors  of  the  corrections  to  the  elements  as  given 
do  not  exceed  a  few  hundredt!>s  of  a  second.  But  the  deter- . 
mmation  of  these  elements  from  observations  of  Mercury  and 
Venus  may  be  seriously  affected  by  the  form  of  the  visible 
disks  of  those  planets,  which  results  in  ol>servations  being 
mada  only  upon  one  limb  when  east  of  the  Sun  and  the  other 
limb  when  west  of  it.  Thus  personal  equation  and  the  uncer- 
tainty of  the  semidiameter  to  be  applied  in  each  I'ase  nuiy  iiave 
an  effect  upon  the  result.  Hut  ])ersonal  e([uation  is  likely  to  be 
smaller  in  the  case  of  Mercury  than  in  that  of  Venus,  owing 
to  the  smallness  of  its  disk. 

There  is  another  circumstance  which  weakens  the  inde- 
pendent determination  of  the  Earth's  eccentricity  and  perigee 
from  observations  of  the  planets.  If  we  define  the  orbit  of  a 
planet,  not  as  a  curve,  but  as  the  totality  of  points  which  the 
planet  occupies  at  a  great  number  of  given  equidistant  moments 
during  its  revolution,  then  it  is  easy  to  see  that  the  general 
mean  effect  of  an  increase  of  the  eccentricity  is  to  displace  the 
entire  orbit  toward  the  point  of  the  celestial  sphere  marked  by 
its  aphelion,  while  the  effect  of  a  change  of  its  perihelion  is  to 
move  the  entire  orbit  u;  its  own  plane  in  a  direction  at  right 
angles  to  the  line  of  apsides.    The  result  is  that  in  a  series  of 


[41',  43 


43,44|  SECULAU  VARIATIONS  OF  THE  SoLAU  ELKMKNTS.         87 


observations  of  ii  i»lsinet  from  the  Earth  the  eonectioiis  to  the 
eccentricity  and  perihehaof  the  two  orbits  can  not  be  entirely 
independent,  and  we  can  determine  with  entire  precision  only 
two  linear  functions  expressive  of  the  relati  ^  displacements 
just  described.  It  may  be  admitted  that,  w  e  observations 
exactly  similar  in  kind  made  around  the  entue  relative  orbit 
in  e(|ual  numbers,  the  etVect  of  the  principle  systemati<'  errors 
would  b(^  n<nirlv  <'liminated  from  the  result.  IJut  we  ciin  not 
rely  upon  this  beiu^'  the  case,  and  «'veii  were  it  the  j-ase  there 
would  probably  ln^  a  residual  etVect  which  would  be  larj^e  in 
proportion  to  the  interdependence  of  the  two  sets  of  correc- 
tions. Hut  in  this  connwtion  thti  important  remark  is  to  be 
made  that,  so  far  as  these  systematic  errors  are  invariable, 
they  would  not  atlect  the  secular  variations,  but  only  the  abso- 
lute values  of  the  elements.  We  may  thercfori^  assif^n  j,'reater 
relative  weights  to  the  tbrmer  than  to  the  latter. 

So  far  as  we  cati  classify  the  results,  I  have  concluded  that 
in  the  <'ase  of  the  secular  variations  of  .f,  e".  and  tt",  the  weight 
of  the  determination  fr<)m  ^[ercury  and  Venus  njight  receivi^  a 
weight  one-tifth  that  from  the  Sun.  But  in  the  ease  of  the 
absolute  values  of  these  ([uantities,  it  would  seem  from  the 
discordance  of  the  results  that  the  relative  weight  of  the 
planetary  results  should  be  muidi  smaller. 

Jn  dealing  with  the  common  error,  a,  of  the  adopted  llight 
Ascensiims  of  the  stars,  it  is  to  be  remarked  that  we  may 
regard  the  observations  in  Itight  Ascension  as  titted  to  give 
the  values  of  a  +  61",  while  rtV"  necessarily  depends  solely 
upon  the  observations  of  de<'lination,  in  effect  if  not  in  form. 
Hence,  although  the  unknown  (|uantities  of  the  solution  are 
a  ami  riV",  I  have  deemed  it  best  to  derive  the  result  by 
regarding  ir  +  61"  as  the  (piantity  to  be  lirst  found,  instead  of 
a  itself. 

Scculay  rariations  of  the  solar  eh'mcnts. 

44.  The  following  table  shows  the  corrections  to  the  tabu)  r 
secular  variations  of  the  solar  elements,  as  they  have  been 
found  from  observations.     In  the  cases  of  Mercury  and  Venus 
the  results  of  b.,th  solutions  are  given  for  the  sake  of  <'0!npari 
son,  although  only  solution  B  is  used.     The  relative  weights 


88 


ELEMENTS  OF  EARTH  S  ORIJIT. 


[44 


have  bet.n  detenu i tied  hy  the  eonsideiations  already  set  forth. 
Ill  the  case  of  Mars,  the  linal  deterininant  of  the  sohition  for 
the  solar  elements  ciiiiie  out  so  nearly  evanescent  as  to  show 
that  no  reliable  values  could  be  obtained,  a  result  whi(!h  we 


Corrections  to  the  secular  varintions  of  the  solar  elements  derired 

from  obserratioHs  only. 


i 

IJ,  rl  e" 

From  observations  of— 

The  Sun        

'■'         70.   !                       '■''         70. 

+0.48    s  '        —0.97     I 
4-0.  27                —0.  58 
-j-o.  39     1           —1.26     I 
l-o.  29                —0.  90 
+0. 32     I           — 1.06     I 
+  1.03    i    

"       70. 

+0.23    5 

-0.47 
+  0.  32       I 
+0.28 
+0.32       I 



Mercury,  solution  A 

IJ 

Venus,  solution      A 

"           B 

Mars 

Mean     . . 

+0.48                —I.  10 

4-0.48                       — I.QO 

+0.26 

4-0.  21 

Adoi)ted . 

^"D,(!t'^      ■    \h[a-\-6l") 

Dta 

From  observations  of — 

The  Sun 

Mercury,  solution  S. 

"              "       B 

Venus,  solution      A 

"          "           B 

Mars 

"         70.                         »         70. 

4-0.32    5          —0.63    2 
— 0.40                — 1.84 
—0.29     I  ,        —2.05     3 

+0-32         ! 

+0.46     I   1         —1.20    2 

// 

+0.34 
— 1.26 
—0.79 

— 0-33 
—0.  14 

Mean 

Adopted 

4-0.25      •            —1.40 
4-0.26                 — 1.30 

—0.  30 
— 0.  30 

might  expect,  because,  in  order  to  separate  the  principal  ele- 
ments of  the  P^arth's  orbit  from  those  of  the  planet,  observa- 
tions should  be  continued  all  around  the  relative  orbit,  whereas, 
as  a  matter  of  fact,  they  are  generally  made  only  near  the  time 
of  opposition.  I  have  judged,  however,  that  the  correction  to 
the  secular  variation  of  the  obliquity  obtained  by  putting 
J^tdl"  =  —  1".00  in  the  eciuat'on  for  DtfJf  might  enter  with 
half  the  weight  that  it  does  in  the  cases  of  Mercury  and 
Venus.  Before  the  final  values  and  weights  of  the  quanti- 
ties in  the  table  had  been  definitively  revised,  provisional 
values  were  used  in  the  subsequent  part  of  the  investigation. 


44,  45]   CORRECTION  TO  THE  STANDARD  DECLINATIONa.  80 

These  i)r()visional  vuliies  are  given  in  the  last  line  of  the  table. 
It  is  also  to  be  noted  that  the  secular  variations  of  i\  c"  rr,  n" 
and  f.  in  the  definitive  tiieory  and  tables  are  those  coniputed 
from  the  adoi)ted  masses  of  the  planets. 

Correction  to  the  standard  of  Ihrlinaiion. 

4."),  The  results  for  the  secular  variati<)n  of  <)",  the  common 
error  of  the  standard  Declinations  witliin  the  zodiacal  limits, 
are  noti  ^iven  in  the  table,  as  other  data  are  available  for  its 
determination.  The  followiiif;  shows  the  separate  values  of 
S  and  its  secular  variation,  derived  from  observations  of  the 
planets  to  Saturn  inclusive.  For  reasons  already  stated  obser- 
vations of  the  Sun  are  not  used  for  this  purpose. 


From  observations  of  Mercury, 

Venus, 
^lars, 
Jupiter, 
Saturn, 


-"  "  tl7. 

(?= -0.18 -0.4!)  T;  2 

-0.19-0.(M>T;  1 

-0.21-0.2:{T;  4 

-0.04-0.4;JT;  3 

+  0.04- 0.08  T;  4 


Mean ;       rJ  =  -  0".09  -  0".42  T 
Adopted;  (S  =  -  0  .08  -  0  .501 


-o.  30 
-0.30 


Not  only  observations  of  the  planets  but  those  of  the  fixed 
stars  are  available  for  the  determination  of  S  and  of  its  secular 
variation.  In  the  discussion  of  the  Declinations  derived  from 
observations  with  the  (xreenwich  and  Washington  transit  cir- 
cles {Atitronomical  Papern,  Vol.  II),  I  have  shown  that  the 
Greenwich  observations  iu.iicate,  with  some  uncertainty,  a 
secular  variation  of  the  corrections  to  the  standard  declina- 
tions which  will  give  a  value  of  about  — 0".o5  for  the  seen 
lar  variation  of  rf.  But  Bradley's  Declinations,  as  reduced 
by  AuwERS,  would  give  a  still  larger  negative  vsilue,  approxi- 
mating to  an  entire  second.  As  the  value  which  we  may 
assume  for  6  does  not  greatly  influence  the  other  elements, 
I  have  adopted  as  a  convenient  probable  result,  the  varia- 
tion -0".50  T. 


flO 


KLHMKNTa  Ol'   EARTirs  OIIIMT. 


f4(} 


Hi  * 


1 1  '■: 


SP 


ii 


If 


IhJinitiniiec'Klar  mriationH  of  the  planvtarij  vie  men  In  from  ohaer- 

iHifions  alone. 

Hi.  Iliiviii^'  «U'(;i(l»Ml  upon  tlic  adoptiMl  valucH  of  tlu-  six 
quiintitios  tonnd  in  tlu^  last  artii'le,  wr  rejjard  theiu  as  known 
(inantitics,  anil  siihstitnfc  tlicni  in  tlie  cliniinatin;;  <M|uations, 
which  }(ive  thv  values  of  tlie  renuiininf''  si'cuhir  variations. 
As  thif  unknown  quantities  in  thesu  4>qniitions  aie  not  tlie 
corrections  themselves,  but  certain  functions  of  thiMU,  we  pre 
pare  the  following;  table,  showinj;  the  formation  of  the  quan- 
tities which  are  to  be  substituted  in  the  several  efpiations. 
The  tal)le  scarcely  seems  to  need  any  explanation,  except  that 
tfia  unknown  diiautities  j^iven  in  the  three  columns  on  the 
rijiht  are  formed  by  dividinj;  the  secuhir  variations  lor  twenty- 
five  years  by  the  coetlicients  {jjiven  in  §  21. 

Ailoptol  secular  variations  of  the  solar  unlcnon'ns,  to  be  substi- 
tutett  ill  the  eliminatinff  equations  for  the  several  planets. 

Mercury.               Venus.  Mars. 

1),  rt7"    =-\".m;  [l"   I'   =-0.2.")();  -(MMUT);  -0.0833; 

l)^6        =-0.50;  [rJ     |'=-0.12r);  -0.(L'.")0;  -0.0250; 

D^n       =-0.30;  [rr    |'   =-0.075;  -0.0750;  -0.0150; 

e"DtrJ7r"=  +  0  .20;  [  .t"  ]'   =+0.108;  +0.03'J5;  -f  0.0325; 

'\(h-"   =+0  .21;  [e"  \'   =  +  0.087;  +0.0210;  +0.0202; 

^rfi      =+0.48:   [^     1'   =+0.120;  +0.0300;  +0.0300. 

To  facilitate  a  Judgment  or  rediscussiou  of  this  part  of  the 
process,  we  give  on  the  next  three  pages  the  normal  eiiuations 
between  all  the  secular  variations  which  renmin  after  the  cor- 
rections to  the  elements  of  the  Sun  and  planets  are  eliminated 
from  the  original  normal  equations.  We  give  these  rather 
than  the  eliminating  ecpiations  which  were  actually  used  in 
the  substitution,  because  they  show  more  fully  the  relations 
betweeu  the  unknown  quantities,  and  can  therefore  be  better 
used  in  any  ulterior  discussion.  Regarding  the  preceding  six 
quantities  as  known,  and  substituting  them  in  the  normal 
equations  for  the  secular  variations,  we  derive  the  detinitive 
values  of  the  secular  variations  which  relate  to  the  planets. 
They  are  shown  in  the  next  table.  In  the  latter  the  values  of 
the  solar  elemen  ts  are  repeated  for  the  sake  of  completeness. 


4<>]        NORMAL  Kl^UATIONS  FOR  SECl'LAR   VARIATIONS.         91 


f? 

^« 

'5 

!t 

•J 

•^ 

1- 

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NOUMAL  Et^UATIONS  FOR  8E0ULAB  VAUIATIONS. 


[46 


^  I  §  i$  s  ^  ^ 

^^     ^       It       U       »       M       I- 


+   I   +  +   I 


?1       CO 


L'     "-<     i-t 


2:1.    O        »H        -H       rZ 


I   +   I    I    I    I 


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40]       NORMAL  EQUATIONS  FoU   SErULAIt   VABIATIONH.         03 

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94 


SECULAR  VABIATIONS  FROM  OBSERVATIONS. 


[46 


•Values  of  the  secular  variations  as  derived  from  observations 

only. 


Uiikuowu.       C'orr.        Tables. 


Result. 


Mercury.       DtC        -.0091  -0.&1  +     4.19  +     3.3G±0.50 

eDtTT      +.1577  +1.30  +116.94  +118.24±0.40 

Dt*        +.0r)93J  +0.83  +     6.31  +     7.14  i  0.80 

siniDt^  +.08iriN  +0.70  -  92.59  -  91.89±0.50 

Dtdl       -.0967  -1.55 

Venus.           Dte        +.1393       +1.67  -  11.13  -     9.46 ±0.20 

eB^TT      +.0685       +0.82  -     0.53  +     0.29±0.20 

Dt*        +.1153  J    -0.65  +     4.52  +     3.87 ±0.30 

siniDt^  -.0592N  -2.73  -102.67  — 105.40±0.12 

Dtrfi      -.1919       -3.84 

Earth.  U^e  +0.21    -     8.76    -     8.55±0.09 

+0.26    +  19.22    +  19.48±0.12 
+0.48    -  47.53    -  47.11  ±0.25 

Mars.  DtC       —.1190       -0.68  +  19.68  +  19.00±0.27 

+0.29  +149.26  +149.55±0.35 

+0.17  -     2.43  -     2.26  ±0.20 

siuiDt^  -.0442N   -0.76  -   71.84  -  72.60±0.20 
J\6l      -.0946       -0.76 

The  first  eohiuiu  of  numbers  in  this  table  gives  the  unknown 
quantity  as  found  immediately  from  the  eliminating  eijuations. 
These  quantities  being  multiplied  by  the  factors  given  in 
§27,  Ave  have  the  corrections  ts)  the  tabuLir  secular  varia- 
tions, as  given  in  the  cclumn  "correction."  The  next  column 
gives  the  value  of  the  tabular  secular  variations,  which  are 
in  all  cases  those  actually  adopted  by  Leverrier.  In  the 
case  of  the  Earth,  as  has  been  pointed  out  by  Sturmer  and  by 
Innes,  the  secular  variation  of  the  radius  vector  does  not  cor- 
respond to  that  of  the  longitude.  l>iit  as  that  of  the  longitude 
is  the  preponderating  quantity  in  its  eft'ect  on  geocentric 


Dte 

eDt  TT 

Dtf 

DtC 

—.1190 

eDt  7r 

+  .0536 

Dti 

+  .1136 

;i  I 


40,  47 


CORKECTIOWS  TO   THE  .SOLAR  ELEMENTS. 


95 


places,  I  luive  regardi'd  the  value  <)f  the  ecceiitr  city  used  in 
the  tables  of  the  equation  of  the  center  as  the  tabular  one  to 
be  adoi)ted. 

The  numbers  in  the  coluum  "  rnknown,"  which  are  followed 
by  the  letttrs  J  and  ^\  are  the  respective  values  of  [.I],  and 
[N|i,  which  are  changed  to  (U  and  siu  i6&  by  the  equations 
of  §41. 

Finally,  we  have  the  results  given  in  the  last  column  for  the 
actual  secular  variations  of  the  several  elements  as  derived 
from  the  preceding  discussion  of  all  the  observations. 

The  result  is  followed  by  the  probable  mean  error  of  each  of 
the  quantities  as  estimated  from  th«  probable  magnitude  of 
the  sources  of  error  to  which  they  are  liable.  As  in  other 
cases,  these  ([uantities  are  very  largelv  a  matter  of  judgment, 
because  the  probable  err«ns  as  determined  in  the  usual  way 
from  the  eliminating  e(|uations  would  be  entirely  unreliable. 

Dcjinitire  corrections  to  the  nolnr  elements  for  18')0. 

47.  Leaving  the  above  results  to  be  subsequently  discussed, 
we  go  on  with  the  solution  of  the  equations.  By  a  continuation 
ux'the  process  Just  described,  we  regard  the  preceding  secular 
variations  as  known  (luantities,  and  substitute  them  in  the 
eliminating  equations  for  the  solar  elements  which  are  derived 
from  the  normal  equations  for  each  planet.  By  this  substitu- 
tion, we  reach  three  fresh  sets  of  values  of  the  corrections  of 
the  solar  elements  themselves,  one  set  from  the  observgtions 
of  each  planet,  which  are  to  be  redm-ed  to  ISoO  and  combined 
with  those  already  found  from  observations  of  the  Sun,  in 
order  to  obtain  the  most  probable  result. 

Here  we  meet  with  the  same  ditliculty  that  confronted  us  in 
the  case  of  the  secular  \  ariations.  With  the  exception  of  the 
Sun's  mean  longitude,  we  are  t«>  regard  the  results  derived 
from  each  of  the  planets  as  uiiected  by  obscure  sources  of 
systematic  error,  the  probab'e  magnitude  of  which  can  only 
be  inferred  from  the  general  deviation  of  the  (piantitios  them- 
selves. As  in  the  former  case,  <*  is  not  regarde<l  as  a  (luantity 
independently  determined,  but  a-\-6i"  has  been  taken  instead. 
The  concluded  value  of  a  is  then  fimnd  by  subtracting  6Vi 
from  61"  4-  a.  Since  the  corrections  to  the  solar  elements 
pertain  to  each  separate  epoch,  those  derived  from  the  obser- 


Si   -: 


f« 


96 


ELEMENTS  OF  EARTH'S  ORBIT. 


[47 


vations  of  the  planets  are  severally  reduced  to  1850,  and  the 
results  are  shown  in  the  following  table : 

Separate  r allien  of  the  corrections  to  the  solar  elements  for  1S50, 
after  the  above  definitive  values  of  the  secular  variations  are 
substituted  in.  the  eliminating  equations  from  solution  B, 
reduced  to  1850. 


From  observations  of- 

The  Sun 

Mercury 

Venus 

Mars  ... 

Adopted  


6l'f 


6e" 


e"&T:" 


It 


— ■  ,lO 

+•13 
+  •13 
+•25 


20 


+•05 
+■07 
-•17 

+•24 


+  .10 

+.48 
+.06 

—  83 


,02 


+.12 


// 
.00 

—•47 
—.07 
—  .82 


a  ^61" 


—  .02 
+  .60 
+  -34 


// 


-  -07 

;-  ^53 
+  -50 


+1. 18  I  +  .94 


— .  04  1     -f-  .  46  I  -j-  .  48 


These  adopted  values  are  eujployed  in  the  subsequent  stages 
of  tlie  discussion,  but  are  not  in  all  cases  regarded  as  definitive. 
In  the  case  of  f  the  value  — 0".20  is  that  which  I  have  actually 
used  in  the  subsequent  determiniitions  of  the  elements,  but  for 
the  final  value  of  the  obliquity  it  will  be  seen  that  I  have 
taken  — 0".15  as  more  probable. 


If 


CHArTER  V. 


MASSES  OF  THE  PLANETS  DERIVED  BY  METHODS  INDE- 
PENDENT OF  THE  SECULAR  VARIATIONS  WITH  THE 
RESULTING  COMPUTED  SECULAR  VARIATIONS. 

48.  The  plan  of  discussion  laid  <lown  in  Chaptor  I  contem- 
plates the  dcterniination  of  the  masses  of  each  of  the  planets 
from  all  data  independent  of  the  secular  variations,  in  order 
to  determine  how  far  the  observed  secular  variations  can  be 
reconciled  with  these  masses.  The  foUowinj;  is  a  summary  of 
these  ileterminations.  The  planets  outside  of  .Jupiter  need  no 
discussion,  as  the  well-known  determinations  of  their  masses 
are  amply  accurate  for  all  our  present  purposes. 

Matts  of  Jupiter. 

X\S.  One  of  the  works  connected  with  the  present  subject  has 
been  the  determination  of  the  mass  of  Jupittr  from  the  motions 
of (ii;3),  Polyhymnia.  My  work  on  this  subject  has  not  ' ;et  been 
printed  in  full,  but  I  have  f;iven  in  Asfronomisrlw  Xathric/ifen, 
No.  3L»4!>  ( Bd.  13(),  S.  130),  a  brief  summary  of  the  results.  The 
mass  of  Jupiter  has  been  derived  not  only  from  the  motions 
of  Polyhymnia,  but  from  such  other  sources  as  seemed  best 
adapted  to  give  a  reliable  result.  The  following  table,  trun- 
scribed  from  the  publication  in  question,  shows  the  sep-irate 
results  and  the  conclusions  finally  reached: 


Wt 


Reciprocal  of  mass  of  Jupiter  from — 
All  observations  of  the  satellites, 
Action  on  Faye's  comet  (Mollek), 
Action  on  Themis  (Kkiteuer), 
Action  on  Saturn  (Hill), 
Action  on  Polyhymnia, 
Action  on  Winnecke's  comet  (v.  Haerdtl),    1047.17     10 


m.  e. 


6690  N  ALM- 


1047.82 

1 

1047. 7'J 

I 

I047.r>4 

.~» 

1047.38 

7 

1(>47.34 

L»0 

1047.17 

10 

1047.35 

1  0.0<i5 

97 

W7 


98 


MASS  OF  JUPITER. 


[40 


'i 


It  will  be  seen  that  the  result  from  observations  of  the  satel- 
lites has  been  assij^ned  a  very  small  weight.  This  course  has 
been  indicated  by  the  circumstances.  Other  conditions  being 
erjual,  tile  greater  the  mass  of  a  planet  the  less  the  propor- 
tionate precision  with  which  that  mass  can  be  determined  by 
observations  on  the  satellites.  In  any  case,  if  the  measures  of 
the  distances  between  the  satellites  and  the  prinuiry  are  in 
error  by  a  small  fraction,  ^t-,  of  their  whole  amount,  then  the 
error  of  the  mass  will  be  in  error  by  the  fraction  3^*-  of  its 
amount.  P'or  reasons  founded  on  the  construction  and  use  of 
the  heliometer,  I  doubt  whether  the  absolute  measures  made 
with  those  forms  of  that  instrument  which  have  been  used  in 
determining  the  mass  of  Jupiter  can  be  relied  upon  within 
their  three-thousandth  part.  If  so,  the  determination  of  the 
mass  of  the  planet  itself  would  be  doubtful  by  its  thousandth 
part  in  each  separate  case.  The  cluuice  of  personal  Cijuatiou 
between  transits  of  the  satellites  and  the  planet  vitiates  in  the 
same  way  the  results  from  observed  transits  of  the  planet  and 
satellites.  Notwithstanding  the  great  retinement  of  the  dis- 
cussion by  Kemi'F  of  observations  made  at  Potsdam,  and  the 
care  with  which  he,  ScriiUK,  and  others  have  determined  the 
mass  of  Jupiter  by  a  discussion  of  all  the  observations  of  the 
satellites,  L  can  not  conceive  that  the  probable  error  of  any 
possible  result  they  cimld  derive  would  be  less  than  0..>  or  0.4 
in  the  denominator. 

In  this  connection  the  discordances  between  the  mass  of 
Saturn,  found  by  Prof.  Hall  and  by  other  observers  from 
ob' ervations  of  the  satellites,  are  worthy  of  consideration. 
They  lead  us  to  suspect  that  perhaps  it  is  through  good  for 
tune  rather  than  by  virtue  of  their  absolute  reliability  that 
determinations  of  the  mass  of  Jupiter  from  observations  of  the 
satellites  have  agreed  so  well. 

As  to  the  weights  assigned  to  the  other  results,  only  the  last 
needs  especial  mention.  Tiie  probable  error  assigned  by  v. 
Haeudtl  to  his  result  is  very  much  smaller  than  that  which 
I  find  for  the  maav  of  all  the  results.  But,  as  remarked  in  the 
paper  in  question,  it  has  received  a  smaller  relative  weight 
than  that  corresponding  to  its  assigned  probable  error,  because 
of  distrust  on  my  part  whether  observations  on  a  comet  can 


[49 


49,50,51]  MASSES  OF  THK  KAUTH,  MARS,  AND  JIPITER. 


90 


be  considered  as  having  always  been  made  on  the  center  of 
gravity  of  a  welldelined  mass,  moving  as  if  that  center  were 
a  material  point  subject  to  tlie  gravitation  of  the  Sun  and 
phiuets.  This  tlistrust  seems  to  be  amply  Justified  by  our 
general  experience  of  the  failure  of  ccunets  to  move  in  exact 
accordance  with  their  ephemerides. 
I  ))ropose  to  accept  tlie  value  thus  found, 

Mass  of  Jupiter  =  1  —  1(>47..'?"» 

as  the  delinitive  one  to  be  used  in  the  i)lanetary  theories. 

Mann  of  Mt(r.s. 

50.  In  consequence  of  the  minuteness  of  the  mass  of  Mars, 
measures  of  its  satellites,  esjuHtially  the  outer  one,  afford  a 
value  of  its  mass  much  better  '^han  can  be  derived  by  its  action 
on  the  planets.  When  nearest  the  earth,  the  major  axis  of  the 
orbit  of  the  outer  satellite  subtends  an  jingle  of  70".  I  can 
not  think  that  the  systematic  error  to  be  feared  in  the  best 
measures,  such  as  those  made  by  Prof.  Hall,  can  be  as  great 
as  lialf  a  second.  It  therefore  appears  to  me  that  the  mean 
error  in  adopting  Prof.  Hall  s  value  of  the  mass  does  not 
exceed  its  tiftietli  part.  This  is  a  degree  of  precision  much 
higher  than  that  of  any  determination  through  the  action  of 
Mars  on  another  jdanet. 

Prof.  Hall's  measures  of  189i»  show  a  minute  increase  of 
the  mean  distance  given  by  his  woik  of  1.S77.     The  result  is — 


,/'"  — 


=  +  0.014 

These  observations,  however,  were  made  when  the  position  of 
the  orbit  of  the  satellite  was  unfavorable  to  an  exact  deter- 
mination of  the  elements  of  motion.  1  have  adhered  to  the 
original  value  in  the  work  of  the  jjresent  chapter. 

Mass  of  the  Earth. 

51.  I  have  already  pointed  out  the  ditticulty  in  the  way  of 
determining  the  mass  of  the  Earth  from  its  action  on  the 
other  planets.  On  the  other  hand,  the  solar  parallax  has,  in 
recent  years,  been  determined  in  various  ways  with  such 
precision  that  the  mass  of  the  Earth  to  be  used  in  the  plan- 


r 


100 


MASS   OF   THE   EARTH. 


[51 


etary  theories  can  best  be  derived  from  it.  The  theory  of  the 
relation  between  the  mass  of  tlie  Earth  and  its  distaiiee  from 
tlie  Sun,  as  <jiven  by  observations  of  the  seconds  pendulum 
an<l  tlie  lenjfth  of  the  sidereal  year,  is  one  of  the  bet-t  estab- 
bshed  results  of  eelestial  nieciumies.  It  is,  in  effect,  the 
principle  on  \vhi(!h  the  lunar  theory  is  constructed.  In  this 
theory  the  disturbinjLj  action  of  the  Sun  is  necessarily  a  func- 
tion of  the  ratio  of  the  mass  of  the  Sun  to  that  of  the  Earth. 
Hut  in  the  aece[)ted  theory  this  ratio  is  eliminated  through 
the  ratio  of  the  lunar  month  to  the  sidereal  year.  From  the 
well-established  ratio  between  the  distance  of  the  Moon  and 
the  len};;th  of  the  seconds  pendulum,  the  ratio  of  the  masses 
of  the  Sun  and  Earth  come  out  of  this  theory  with  jyieat 
l)recision.  It  need  not  be  developed  here;  it  will  suftice  to 
give  the  iiumeri<!al  result,  which  is  that  between  the  ratio  M 
(»f  the  mass  of  the  Sun  to  that  of  the  Earth  and  the  mean 
eijuatorial  horizontal  i)arallax  of  the  Sun  in  seconds  of  arc 
there  exists  the  relation 


n'M  =  18.3.54031 


I  have  derived  seven  values  of  the  solar  parallax  by  ditterent 
methods,  (»f  which  the  following  are  the  preliminary  results: 


(JiLi/s  observations  of  Mars,  1877, 

(>)ntact  observations,  transits  of  Venus. 

Aberration  and  velocity  of  light. 

Parallactic  e(|uation  of  the  Mocm, 

Measures  of  small  planets  on  Gill's  plan,   8.8CV  ±  .007 

Leveruieu's  method,  8.818  ±  .030 

Measures  of  Venus  from  Sun's  center,  8.8.">7  ±  .0132 


8.780  i  .020 
8.704  i  .018 

8.798  1:  .00.") 

8.799  i  .007 


Wt. 
1 

1 

16 

5 

8 

0..5 

1 


Mean  result,  tt  =  8".802  i  0".00o 


; 

f; 

■i 

'. 

- 

1 
I 

1   - 

y 

i. 

i 

L 

1 

: 

1  have  provisionally  taken  this  mean  a.,  the  most  probable 
value  of  the  solar  parallax  derived  from  all  sources  excei)t  the 
m  iss  of  the  Earth.    The  above  relation  then  gives 

M  =  332  040 


f51 


r»i,r»2| 


MASS   OF  VENTS. 


101 


Taking  for  the  mass  of  the  Moon  1  -i-  <Sl.r»L',  we  have  for  the 
ratio  of  the  combined  masses  of  tlie  Earth  and  Moon  to  tlie 
mas8  of  the  Sun 

vi"  =       ^ 

3i',S  OK) 

a  result  of  which  the  i>robal>U'  error  may  be  rejjfarded  as  some- 
tiiing  more  tlnin  ii  thousandth  i)art  of  its  whole  amount. 

Mann  of  Venus. 

r>L*.  The  mass  of  Venus  adopted  in  the  provisional  theory, 
to  which  Levkkrieu's  tables  were  redm-ed,  was  .000  002  4885 
=  1  -^  401847,  which  is  tiiat  of  Leveurier's  tables  of  Mer- 
cury. In  the  precedinj;  discussicuis  the  foHowing'  three  factors 
of  ciurection  to  this  mass  have  been  found: 


From  observations  of  the  Sun 
From  observations  of  Mercury 
From  obs<'rvatious  of  Mars 

Mean 


—  .0118  J  .0034 

-  .0121  4  .(K):>0 

—  .007«i  :1  .  (  ?  ) 

-  .0119  i  .0028 


rent 


1 
L(> 
5 
8 
0.5 


The  mean  error  assigned  to  the  result  from  observations  of 
the  Sun  may  be  regarded  as  real,  because  the  result  is  the 
mean  of  a  great  nund^er  of  completely  independent  <letermina 
tions,  among  which  no  common  error  is  either  a  priori  prob- 
able or  shown  by  the  discordance  of  the  results.  In  the 
case  of  Mercury,  how«»ver,  as  already  rennirked,  the  effect  of 
systenmtic  errors  is  such  that,  altlu>ugh  they  are  almost  com- 
pletely eliminated  from  the  result,  the  mean  error  computed 
in  the  usual  way  would  be  misleading.  The  weight  assigned 
is  therefore  Is. -gely  a  matter  of  Judgment. 

The  fact  that  it  was  necessary  to  introduce  an  empirical 
correction,  with  a  period  of  about  forty  years,  into  the  n»eau 
longitude  of  Mars,  vitiates  the  deternjination  of  the  nmss  of 
Venus  from  its  action  on  that  planet,  because  one  of  the  prin- 
cipal terms  in  the  action  of  Venus  on  Mars  has  a  period  which 
does  not  differ  from  forty  years  enough  to  make  the  determi 
uatiou  of  the  mass  independent  of  this  empirical  correction. 
I  have  therefore  assigned  no  weight  to  the  result.    We  thus 


T 


102 


MASS  OF  MERCURY. 


152,53 


I 


have  for  the  masa  of  Venus,  as  derived  from  tlie  periodic  per- 
turbations of  Mercury  and  the  Earth  produced  by  its  action. 

!»'  =  1  -j-  lOfi  cm  i  1140 

Mass  of  Mercurif. 

03.  The  mass  of  Mercury  which  I  have  heretofore  adopted, 
1-^7  500  000,  was  rather  a  result  of  {general  estimate  than  of 
exact  computation.  The  fact  is  that  the  determinations  of 
this  mass  have  been  so  discordant,  and  varied  so  much  with 
the  method  of  discussion  adopted,  that  it  is  scarcely  possible 
to  fix  upon  any  definite  number  as  expressive  of  the  mass. 
An  examination  of  Leverrier's  tables  of  Venus  shows  that 
with  the  mass  of  Mercury  there  adopted  (1:3  000  000)  Mercury 
freciuently  produces  a  perturbation  of  more  than  one  second 
in  the  lieliocentric;  longitude  of  Venus.  When  the  latter  is 
near  inferior  conjunction,  the  a(;tual  perturbation  will  be  more 
than  doubled  in  the  geocentric  i)lace,  so  that  the  latter  might 
not  infrequently  be  changed  by  1",  even  if  the  mass  of  Mer- 
cury be  less  than  one-half  Leverrier's  value.  It  was  there- 
fore to  be  expected  that  a  fairly  reliable  value  of  the  mass  of 
Mercury  wouUl  be  obtained  from  the  periodic  perturbations 
of  Venus. 

IJcferring  to  §  -7,  it  will  be  seen  that  the  indeterminate  nmss 
of  Mercury  appears  in  the  equations  in  the  form 

1  +  7//    - 
3  000  000 

From  the  solution  B,  §  38,  the  value  of  jn  comes  out 

/<  =  -  0.0834 

corresponding  to  a  mass  of  INIercury  of  1:7  210  000.  But  in 
a  subsequent  solution  of  the  equations,  when  the  secular  vari- 
ations are  determined  from  theory  and  substituted  in  the 
normal  eijuation  for  /j,  we  find 

yu  =  -  0.0889 
which  gives 

m  =  1  -j-  7  943  000 

The  work  of  the  present  chapter  is  based  on  the  former 
value. 


m 


u 


08] 


MASS  OF  MEKCTRY. 


103 


A  consideration  of  the  inol)jibIe  error  of  tliis  rosult  in  inipor- 
ttiiit.  The  fortuitous  errors  which  mostly  affect  it  are  of  the 
chiss  whieli  I  have  termed  semi-Hi/Htemofic.  Under  this  term  I 
itn'Uide  tliat  larjje  class  of  errors  which,  extendinj;  through  or 
injuriously  atfeciinji  a  limited  series  of  observations,  cause  the 
probable  error  of  a  result  to  be  larfjer  than  that  ^nven  by  the 
solution  of  the  e<iuations,  but  which,  nevertheless,  like  purely 
accidental  ones,  wotdd  be  eliminated  from  the  mean  result  of 
an  infinite  series  of  observations.  To  this  class  belonj;  the 
eiTors  arisiuj;  from  jtersonal  equation  in  observing  the  limb  of 
Venus,  oi,  what  is  the  same  thiu}''.  a  dittereiice  between  the 
practical  semidiauu^ter  corresponding^  to  the  observer  and  that 
adopted  in  the  reductions.  We  may  suppose  that,  during  a 
period  of  several  days,  when  Venus  is  not  far  from  inferior 
(conjunction,  its  geocentric  position  is  atlected  by  a  pertnrba- 
ti(m  produced  by  Mercury.  Thr()uj;h  the  error  alluded  to,  ail 
the  observations  made  by  any  one  observer,  and  in  fa«'t  all 
that  are  made  anywhere,  may  be  alVectj'ii  by  a  certain  con- 
stant error  in  IJight  Ascension.  Near  another  inferior  con- 
Juiu'tion  the  san»e  state  of  things  may  be  rei>eati'd,  with  the 
perturbation  in  the  opposite  direction.  If,  now,  the  observa- 
tions were  nnide  by  the  same  observer,  and  under  the  same 
circumstances,  the  personal  error  would  be  eliminate<l  from 
the  mean  of  tliese  two  results  so  far  as  the  mass  of  Mercury  is 
concerned.  But  very  frequently  ditt'erent  observers  will  h.ave 
made  the  observati(ms  under  the  two  circumstances,  and  dif- 
ferent conditions  will  have  prevailed.  Thus,  it  is  only  throufjh 
the  general  law  of  averages  that  we  can  exi)e(!t  the  eH'ect  of 
these  fortuitous  but  systematic  errors  to  be  completely  elim- 
inated. That  they  would  be  eliminated  in  the  long  run  is 
evident  from  the  fact  that  there  can  be  no  permanent  rehi- 
tion  between  the  personal  equations  of  the  observers  and  the 
changes  in  the  action  of  ^[ercury  upon  Venus.  M«)reover, 
Venus  has  been  observed  with  a  fair  degree  of  ac<Miracy 
through  more  than  half  a  century,  and  it  seems  reasonable 
to  suppose  that  during  that  time  the  ernms  in  question  would 
nearly  disappear. 

It  is  clear  from   these  considerations   that   the   probable 
error  derived  from  the  solution  of  the  ecpiations  would  be 


S!  ' 


104 


MASS  OF  MEIU'LKY. 


[53 


entirely  miHluii(liiif>:.  Hut  a  probable  error  which  ought  to  be 
reliable  can  be  obtaiued  by  a  proj-ews  similar  to  that  which  1 
have  adoi)ted  elsewhere  in  this  paper,  namely,  dividing  up  the 
materials  into  periods,  and  <letermining  the  probable  error  from 
the  discordances  among'  the  results  of  the  several  periods. 
This  probable  error  will  be  reliable,  because  there  is  no  reason 
why  the  same  error  should  affect  the  mass  of  Mercury  through 
any  two  periods.  I  therefore  take  the  partial  normal  ecjua- 
tions  in  fx  derived  from  Right  Ascensions  during  the  several 
periods,  substitute  in  them  the  values  of  the  unknown  quanti- 
ties found  from  solution  Ji,  //  excepted,  and  thus  form  six- 
teen i)artial  normal  equations  in  fA.  These  equations  may  be 
changed  into  the  corresponding  ecjuations  of  condition,  of 
weight  unity,  by  dividing  each  by  the  square  root  of  the 
coefficient  of  the  unknown  (luantity.  The  residuals  then  left 
when  the  definitive  value  of  the  unknown  <iuantity  is  substi- 
tuted will  be  those  from  whose  discordance  the  probable  error 
may  bo  inferred. 

The  partial  normal  equations  thus  found  from  the  Bight 
Ascensions  are  as  follow: 


^ 


17r.(M(52. 

44// 

=  -  38 

183()_'40. 

5649  /u  = 

-  831 

17U5-74. 

12(M 

—  1(15 

1840-'49. 

2913 

-   18 

1775- 8(i. 

1.") 

—   5 

184l)-'5(). 

2238 

-  49 

1787-'i>G. 

20l> 

+  53 

1857-'(i4. 

4506 

-  129 

17!MU'0«). 

345 

+  11) 

1865-71. 

7736 

-  265 

180(;-'14. 

431) 

4-  135 

1871-'79. 

70(52 

-  761 

18U-'L5). 

1)42 

+   2 

1879-86. 

4958 

-  407 

1820-'30. 

178(5 

-330 

1885-'92. 

9561 

-1306 

Sum :     49  (5(58  //  =  —  4095 

^=-  0.0824  ±  .019 

The  difference  between  this  value  of  ju,  which  is  obtained 
only  to  find  the  probable  error,  and  that  formerly  found,  arises 
principally  from  the  omission  of  the  declination  equations. 
The  mean  error  corresponding  to  weight  unity  comes  out 

£1  =  i  4".2 


US]  MASS  OF   MEKCVRV.  l(>r> 

>vhich,  us  aiitieipiitcd,  ih  niiicli  larg<>r  than  that  which  woiiUl 
be  given  by  the  iliscordance  of  the  <>rij;iiinl  observations. 
Tliis  does  not  mean  that  the  original  observations  are  atVected 
by  any  mucIi  mean  t-rror  as  A:  4'MJ,  but  that  the  tliseordances 
between  tlie  10  values  of  /*  are  as  great  as  w»'  hIiouM  rxpeet 
them  to  be  if  the  origiiuil  observaticms  were  absohitely  free 
from  systematie  error,  but  atfeeted  by  purely  aceidental  eriors 
uf  this  mean  amount. 

The  results  ot'  tin*  suhition  for  the  mass  of  Mercury  nuiy  be 
expressed  in  the  form 

1  i  0..*i2  ,    1  ±  0.35 


7  210  000 


7  i»  l.{  000 


In  all  researches  which  have  been  nnide  on  the  nioti(»n  of 
Encke's  comet  by  Hncke,  von  Asten,  and  IJACKLr.MJ,  the 
determination  of  this  mass  has  been  kept  in  view.  The 
results  are,  liowever,  so  discordant  that,  as  already  rcmarkcl, 
scarcely  any  definitive  result  can  be  derived  from  them. 

To  this  statement  there  is,  however,  one  apparent  execption. 
Ill  an  appendix  to  his  very  careful  and  elaborate  discussion  of 
Winnecke's  comet,  vox  Haeudtl  has  derived  the  value  of 
the  mass  of  Mercury  from  all  the  return  of  Encke's  conu't  as 
worked  up  by  voN  AsjTEN  and  Hacklund.*  The  only  inter- 
pretation which  1  can  put  upon  hia  result  is  this:  If  we  regard 
the  acceleration  of  the  c<nnet,  which  it  is  supposed  results 
from  all  the  observations  made  upon  it,  as  non-existent,  the 
following  two  nuisses  of  Mercury  are  derivable  from  the  obser- 
vations: 

181«>-I8«J8,  Ml  =  1  4-  .")  (i  18  (KM)  ±  2000 
1871-1885,  w  =  1  -r  o  009  700  ±  000  000 

He  also  finds,  from  the  motion  of  Winnecke's  comet, 
wj  =  1  -1- .-)  012  842  ±  Ol>7  803 


*  Denkschrifteii  der  Kaiaerlichen  Akademie   der  Wisseuschaften,  Vol. 
56,  p.  172-175.     Vienna,  1889. 


; 


100  MASH  OF   MEUCl'RY. 

and  from  four  oquatioiis  of  Leverkikk 


1 53, 54 


1  4. .-» .->14  700  4  m)  (MM) 


III 


Tlu'  consistnM'y  of  tlieso  results  seems  to  me  entirely  Weyond 
what  the  observjitions  art'  capable  of  giving,  and  I  hesitate  to 
ascuihe  great  weight  to  them.  Moreover,  the  result  implitiitly 
contained  in  these  numbers,  that  tlie  supposed  He<;ula]-  accel- 
eration of  the,  comet  <lisappcar8  when  we  attribute  the  pre- 
ceding mass  to  \r«'rcury,  merits  fartlier  inquiry. 

The  probable  density  of  the  ]>lanet  may  form  a  basis  for  at 
least  a  rude  estimate  of  its  probable  mass.  The  faet  that  the 
Kaith,  N'enus,  and  Mars  have  <leu8ities  not  very  different  from 
eaeh  otlier,  while  that  of  the  Moon  is  HM  the  density  of  the 
Earth,  leads  us  tn  suppose  that  Mercury,  being  nearest  to  the 
Moon  in  mass,  has  probably  a  slightly  greater  density.  Its 
diameter  at  distance  unity  has  been  repeatedly  measured  and 
found  to  bo  <l".0,  or,  roughly  speaking,  three-eighths  that  of  the 
Earth.  Were  its  density  0.7,  its  mass  would  therefore  be 
about  1  :  !>,(MM),000.  In  view  of  the  fact  that  the  measured 
diameter  is  probably  somewhat  too  small,  these  consider- 
ations lead  us  to  conclude  that  the  uuiss  is  probably  between 
1:0,000,000  and  1:0.000,000. 

As  the  v.alue  of  the  mass  to  be  used  in  investigating  the 
secular  variations,  I  have  adoi)ted 


y  =  +  0.08 


Mass  of  Mercury  = 


1.08 


7  500  000 


Secular  variations  resulting  from  theory. 

54.  In  the  Astronomical  Papers,  Vol.  V,  Part  IV,  were  com- 
puted the  secular  variatious  of  the  elements  of  the  orbits  in 
question  using,  as  the  basis  of  the  work,  the  values  of  tlie 


\fc_:..,-^ 


ri4i 


TiiKoiMrrrcvL  skculau  variations. 


to; 


iiiiiHs»'s  wliosr  leoiprociils  iin-  toiiinl  in  tlir  colmnii  A  1h»|(>\v. 
Ill  (.'oliimu  U  aro  cited  the  musses  whicU  I  have  decided  upon. 


A 

B 

OriKiual 

AdpohMl 

rociproiiil 

rucipriMiil 

ot'  mass. 

of  muss. 

y 

Mercury, 

7."»(KMMM) 

<;  on  14 1 

+  .O.S0 

Wmiiis, 

no  <)()(» 

KM  J  7.'»0 

4-.<M»f>0 

Kurtli  +  Moon, 

;L'7  000 

.i'JS  000 

-.oo;{or> 

Mills, 

.^^^{^(N) 

.iO!);(.joo 

0 

Jupiter, 

104  7. S8 

io47..{r) 

4-.ooo.*)<» 

Saturn, 

;{5oi.<i 

0 

Uranus, 

L*L'  7."»<; 

0 

Neptune, 

10  r.  10 

0 

In  the  case  of  the  I-jarth  we  iiave  to  add  tlie  sccMihir  varia- 
tion of  the  ])erihelion  prodiu't'd  hv  the  non  spluMiiuty  of  the 
system  Karth  +  Moon.     For  the  juincipal  term  I  have  found, 

D,e"fJ,T"  =  +  O'MiiO 

The  resultinj?  vahies  of  the  secuhir  variations,  expressed  as 
functions  of  /-,  r',  r",  i'"',  are  given  in  the  following'  section: 

Theoretical  secular  niriatioiiHf'or  is'iO. 
McrcKri/. 


Dte       =+     4.22  4-0.<><>'^+  --Sc'-f   1.1 /'"-O.l ('"'  =  +     4.21 

el)t;r,     =  +  100.3(»  4-0.00   -|-r.(J.8    4-18.8     4-O.r)       =4-100.70 

Dt/       =-|-     0.7(1  -0.04    -  0.0    -  1.4     4-0.0       =4-     0.70 

sin  /  D, '^  =-  02.12  —0.3;{    -4!>.;i     -12.2      —1.2       =-  02..j0 

Veil  lift. 

I>tC       =-     O.r.8  -1.30J/4-  O.Oj''-  4.0i'"— 0.2/'"'=-     0.(»7 

eDtar,     =-f     0.39-0.81   4-0.0    -  3.0     4-0..")       =-|-     0.34 

Dtt       =-f     3.43  4-0.70   -\-  0.0    -f  0.0     —0.3      =+     3.49 

sin  iDt^  =-105.92  4-0.20   —20.2    —43.2      -1.2       =-10<;.00 


108 


el),  TT 


THEORETICAL  SECULAR   VARIATIONS 
Uarfh. 

:-     8..")7  _o.lL>//_|_   i.;i,,/ 
=  +   llUd  -0.18   ^   5  ,s 
:-   4(}.<M  _0.i>l    _28.3 


[54 


-1.0,'"'=  _        ^.^7 

+  1.0      =+  1JI..39 
-0.7       =_  4(j.80 


Mars. 


--+   18.71  +0.(13,-+  0.1,.'+  -J,!,."     "        _  ,    ,0';, 
=  +  U8.8.+0.0«  +  4.«+,.,,  Z+'l;^ 

—     .'..«  -0.04  +1L>.0    +0.0     +0.0,."'=_     ..o, 
-   72.43 -0.27   -.".,1    _;.4     I,.,      ^_     - 


i 


CHAPTER  Vr. 

EXAMINATION  OF  THE  HYPOTHESES  BY  WHICH  THE 
DEVIATIONS  OF  THE  SECULAR  VARIATIONS  FROM 
THEIR  THEORETICAL  VALUES   MAY  BE   EXPLAINED. 

oo.  Tlio  inve8tijj:ationH  of  the  present  cliapter  are  founded 
on  a  comparison  of  the  secular  variations  derived  purely  from 
observations  in  Chapter  IV,  with  those  resultinj;-  from  the 
values  of  the  masses  obtained  independently  of  the  secular 
variations  in  the  last  chapter.  For  the  sake  of  clearness, 
these  two  sets  of  secular  variations  and  their  dilferenccs  are 
collected  in  the  following  table.  The  mean  errors  assigned  to 
tlu^  theoretical  values  are  those  which  result  frotn  the  prob- 
aWe  mean  errors  of  the  respective  masses.  They  are  there- 
fore not  to  be  regarded  as  independent.  The  mean  errors 
given  in  the  column  of  differences  are  those  which  result  from 
•I  combination  of  those  of  the  other  two  colunms.  The  errors 
of  the  observed  quantities  must  not,  however,  be  judged  from 
those  of  the  ditlorences,  because  subseqiieut  changes  in  the 
masses  of  Mercury,  Venus,  and  the  Earth  nuty  produce  a 
general  diminution  in  the  discordances. 

Mercury. 


Observation. 
//  // 


T'.R'ory. 
//         // 


Diff. 


\/w. 


Dte       +     3..%i 0.50-1-     4.24^.01  -0.88-1  . no  -0.80  :i 

c'DtTT      +118.1'4  L0.40 -f-10!>.70  t.lO -f8.18L.43     .     .  0 

Dt»       +     7.14  4,0.80-1-     0.704-.01 -f-0..'J8:t-.80 -f0.;{8  1^ 

sin  iDifi  -  Ol.80i0.45  _  0L>..->0i:.10  -f0.01i..5l>  -fO.23  2.2 

Venus. 

Dte       -     0.40±0.20-     0.07i.24 -|-0.21±.31  -f  0.12  5 

el\7c      -f     0.20i:0.20 -f-     0.34i.l5  -0.05±.25     .     .  0 

Dti       -f     3.87±0.30-|-     3.40 ±.14 +0.38 i. 33  4-0.44  3^ 

siniD^e  -10.5.40±0.12  -100.{K)i.l2  +0.fiOi.l7  +0.52  8 

109 


110 


COMPARISON   or   SEC;ULA11  VARIATIONS.  [55 

Earth. 
ObBervation.  Theory.  Ditf.  J        y,r. 


//        /' 


DtC        -     8.r)54(>.(M>  -     .S.57i.()4  4-0.02i.10  +0.0L'    10 
eDtTT      +  19.48-i:0.1li  +   10.38i.05 +0.10i.l.'i     .    . 
Dtf       -  47.11  i0.2;{  -  40.S!>±.01>  -0.22±.27  -0.40      4^ 

Mars. 

l\e       -\-   19.004:0.27  4-   18.71±.01  -|-0.20±.L>7  4-0.29  ;i.7 

eDt^r      + 149..").")  4- 0.35 +148.804.04 +0.7.")  4. .T)    .     .  0 

Dj/       -     2.2(J40.20  -     2.254.04-0.014.20+0.08  5 

siiiiDt'^  -   72.0040.20  -  T2.034.09  +0.034.22  -0.17  5 

If  we  umltiply  the  nieuu  errors  given  by  0.0745,  to  reduce 
tliem  to  probable  errors,  we  shall  see  that  only  four  of  the 
fifteen  ditterences  are  less  than  their  probable  errors.  The 
deviations  which  call  for  especial  consideration  are  the  follow- 
ing four : 

1.  The  motion  ()f  the  perihelion  of  Mercury.  The  discord- 
ance i'.i  che  secular  motion  of  this  element  is  well  known. 

2.  The  motion  of  the  node  of  Venus.  Here  the  discordance 
is  more  than  five  times  its  probable  error. 

3.  The  perihelion  of  Mars.  Here  the  discordance  is  three 
times  its  probable  error. 

4.  The  eccentricity  of  ]\Iercury.  The  discordance  is  more 
than  twice  its  probable  error.  It  is  to  be  emarked,  however, 
that  the  ])robrble  error  of  this  quantity  is  very  largely  a 
matter  of  judgment,  and  that  its  value  may  have  been  under- 
estim.'ted. 

The  deviations,  if  not  due  to  erroneous  masses,  may  be 
explained  on  two  hypotheses.  One  is  that  propounded  by 
l*rof.  Hall,*  that  the  gravitation  of  the  8un  is  not  exactly  as 
the  inverse  square,  but  that  the  exponent  of  the  distance  is  a 
fraction  greater  than  2  by  a  certain  minute  constant.  This 
hypothesis  accounts  only  for  the  motions  of  the  perihelia,  and 
not  for  any  other  discordances. 

The  other  hy])othesi8  is  that  of  the  action  of  unknown 
masses  or  arrangements  of  matter.    Since  the  latter  hypothesis 

* AaH'onomical  Journal,  Vol.  XIV,  p.  7. 


li 


55, 56J 


NON-SPHERICITY   OF  THE  SUN. 


HI 


would  account  for  other  motions  than  those  of  the  perihelia,  it 
might  seem  that  the  existence  of  the  other  discordaniies 
tells  very  strongly  in  its  favor.  The  hypotheses  of  possible  dis- 
tributions of  unknown  niatter,  therefore,  have  tirst  to  be  con- 
sidered.* 

Hypothesis  of  nonsphericity  of  the  Sun. 

56.  In  a  case  where  our  ignorance  is  complete,  all  hypotheses 
which  do  not  violate  known  facts  are  admissible.  Beginning 
at  the  <;enter  and  passing  outward,  the  lirst  question  arises 
wh<  ther  the  action  may  not  be  due  to  a  non-spherical  distri- 
bution of  matter  within  the  body  of  the  Sun,  resulting  in  an 
excess  of  its  polar  over  its  equatorial  moment  of  inertia.  The 
theory  of  the  Sun  which  has  in  recent  times  been  most  gener- 
ally accepte<l  is  that  its  interior  nniy  be  regardi'd  as  gaseous, 
or  rather  as  a  form  of  matter  which  combines  tiie  elasti«'ity 
and  mobility  of  a  gas  with  the  density  of  a  li«iuid.  Such 
being  the  case,  we  may  conceive  that  vortices  of  which  the 
axes  coincide  with  that  of  rotation  may  exist  in  the  interior 
in  sufih  a  way  that  the  surfaces  of  equal  density  are  non- 
spherical.  A  veiy  small  inecjuality  of  this  sort  would  suflice 
to  account  for  the  motion  of  the  perihelion  of  .Arercury, 

This  hypothesis  admits  of  an  easy  test.  Whatever  be  the 
nature  or  amount  of  the  inecpiality,  a  simple  computation 
shows  that  to  account  for  the  observed  i»henomenon  it  is 
necessary  and  sufficient  that  tlie  e<iuipotential  surfaces  at  the 
surface  of  the  Sun  should  have  an  ellipticity  of  r.ather  more 
than  half  a  second  of  arc.  It  can  not,  I  conceive,  be  doubted 
that  the  visible  photosphere  is  an  e([uipotential  surface.  We 
have  then  to  inquire  whether  th«'re  is  any  such  ellipticity  of 
the  photospliere  as  that  rcijuired  by  the  hypothesis.  This 
question  seems  completely  set  at  rest  by  the  great  mass  of 
heliometer  measures  made  by  the  (lerman  observers  in  con- 
nection with  the  transits  of  Venus  of  1874  and  1S82,  which 
have  been  discussed  by  Dr.  Auweus.     The  general  result  is 


*Aftcr  carrying  out  tlio  investigations  of  this  chaiitcr,  I  find  that  the 
Biibjcct  was  stiidiod  on  Niuiilar  lines  by  Dr.  P.  Hau/iu  iieiuly  tlireti  y«;iir8 
ago,  and  that  I  made  certain  suggestions  on  the  Hubject  to  Dr.  Bacsch- 
iNdEii  ten  years  ago.  See  Astvononiixhr  Xavhrichten,  Vol.  109,  p.  32,  and 
Vol.  127,  )..  81. 


112 


INTRA-MERCURIAL  GROUP. 


[56,57 


!;l.i.- 


that  the  mean  of  the  equatorial  measures  are  slightly  less  than 
the  mean  of  the  polar  measures,  the  difference,  however,  being 
within  the  probable  errors  of  the  results.  I  conclude  that 
there  can  be  no  such  nonsymmetrical  distribution  of  matter 
in  the  interior  of  thcs  Sun  as  would  i:roduce  the  observed  effect. 
This  same  conclusion  seems  to  apply  to  matter  immediately 
around  the  photosiihere,  Au  equatorial  ring  of  planetoids,  or 
gaseous  substances  of  the  required  nuiss,  very  near  the  photo- 
sphere, would  render  the  equipotential  surfaces  of  the  photo- 
sphere elliptical  to  a  degree  which  seems  precluded  by  the 
measures  in  question.  At  a  very  short  distance  from  tlie  sur- 
face, however,  the  effect  would  be  inappreciable. 

Hfipothcsia  of  an  intra-mc'cnrial  ring  or  {/roup  of  planeioMa. 

57.  Passing  outward,  we  \\AsG  next  to  consider  the  hypothe- 
sis of  an  intra-niercurial  ring  adequate  to  produce  the  observed 
phenomena.  In  a  lirst  approximation  we  niay  suppose  the 
ring  circular.  Its  mass  can  not  be  determined,  because  it  will 
depend  upon  the  distance;  we  have  to  determine  a  certain 
function  of  the  mass  and  distance  adequate  to  produce  the 
observed  motion  of  the  perihelion.  Then  we  must  inquire  what 
effect  the  ring  will  have  on  the  secular  variations  of  the  other 
elements,  both  of  Mercury  and  of  the  otaer  planets,  and  see  if 
these  effects  can  be  reconciled  with  observation.  In  the  com- 
putations I  have  assigned  to  the  excess  of  motion  the  pro- 
visional value  40". 7.  If  the  ring  is  not  very  distant  from  the 
Sun  the  motion  which  it  will  produce  in  the  perihelion  of  a 
planet  whose  mean  motion  is  n  and  whose  mean  distance  is  a 
may  be  represented  in  the  form 


-a 


a' 


1.1  being  a  function  of  the  mass  of  the  ring  and  of  its  radius, 
which  is  nearly  the  same  for  all  of  the  planets,  so  long  as  the 
radius  of  the  ring  is  only  a  small  fraction  of  the  distance  of 
Mercury.    A  first  approximation  to  ja  is — 


4 


571 


INTRA-MERCrRIAL   GROUP. 


iia 


m  being  the  ratio  of  its  mass  to  that  of  the  Sun  iintl  r  its  radius. 
Multiplying  these  motions  in  tlie  case  of  the  four  planets  by 
their  eccentrii-ities,  we  lin<l  that  the  hypothetical  ring  will 
produce  the  following  secular  variations: 


>Ier<'ury,   1),  tt 
Venus, 
Earth, 
Mars, 


40.7;  «  Dt  7T  =  8.;W 
4.6  {}.0M 

1..5  0.025 

o.;j4  o.o;u 


Owing  to  the  sniallness  of  the  eccentricities  the  effect  is 
insensible,  except  in  the  case  of  Mercury,  so  tliat  the  ring  will 
not  account  for  the  observed  excess  of  motion  of  the  perihelion 
of  ]Mars. 

Such  a  ring  will  necessarily  ])roduce  a  motion  of  the  plane 
of  the  orbit  of  Mercury  or  Venus,  or  of  both,  because  it  can 
not  lie  in  the  plane  of  both  orb;ts. 

Let  us  put  ii  for  its  inclination  to  the  ecliptiit,  and  ^i  for  the 
longitude  of  its  node  on  the  ecliptic;  and  let  us  put,  also, 

pi  =  /[  sin  ^1 
f/i  =  /,  cos  ^1 

and  let  y>, />',  .  .  .  ,  r/,  </',  .  .  be  the  corresponding  (juan 
titles  for  the  i)lanets.  The  theory  of  the  secular  variations 
then  shows  that  the  ring  will  produce  a  motion  of  the  plane  of 
the  orbit  of  Mercury  given  by  the  e<|uations 


II  n 


^^ti^'='  "('Vi-'i)  =  40".7iv, -</) 


l^t'yi  =  ^^f{i>-i?,)  =  40".7(i>-^,) 


Expressing  the  motions  of  p  and  q  in  terms  of  the  motions  oft 
and  ^,  which  is  necessary,  owing  to  the  very  dilferent  weights 
of  the  determination  of  the  motion  of  the  planes  of  Mercury 
and  Venus  in  the  dire<;tion  of  these  two  coordimites,  we  have 
5<)90  N    ALM 8 


114 


INTRA-MERCrRIAL  GROUP. 


[57 


I  i'i 


5*!   '' 


;:l 


I 


the  following  expressions  for  these  two  motions,  which  we 
equate  to  the  observed  excesses  :* 


// 


4.90  +  26.9  qi  +  2SApi 
0.27+    0.8      +    3.0 
0.00  +  28.4      -  26.9 
0.00+    3.0      -    0.8 
0.00        0.0     —    1.5 


+  0.57  ±  0.50 
+  0.63  ±  0.12 
+  0.50  ±  0.80 
:  +  0.45  ±  0.30 
-  0.25  ±  0.25 


Multiplying  the  conditional  ejiuations  thus  formed  by  such 
factors  as  will  make  the  mean  error  of  each  equation  nearly 
dL  0".5,  we  have  the  following  conditional  eciuations  for  pi 
and  9i : 


27(/, 

+  2Spi 

// 
=  +  5.53 

3 

+  12 

=  +  3.00 

17 

-16 

==  +  0.30 

5 

-    1 

=  +  0.77 

0 

-    3 

=  -  0.50 

The  solution  of  these  equations  gives  very  nearly 


9i  =  +  0.11; 
i)i  =  +0.12; 


?,  =  9° 
^1  =  48 


This  great  inclination  seems  in  the  highest  degree  improbable 
if  not  mechanically  impossible,'  since  there  would  be  a  tend- 
ency for  the  planes  of  the  orbits  of  a  ring  of  planets  so 
situated  to  scatter  themselves  around  a  plane  somewhere 
between  that  of  the  orbit  of  Mercury  and  that  of  the  invari- 
able plane  of  the  planetary  system,  which  is  nearly  the  same 
as  that  of  the  orbit  of  Jupiter.  Moreover,  the  motion  of  the 
l)erihelion  of  Mars  is  still  unaccounted  for  and  that  of  the 
node  of  Venus  only  partially  accounted  for,  as  shown  by  the 
large  residual  of  the  second  equation.  In  fact,  the  great  incli- 
nation assigned  to  the  ring' comes  from  the  necessity  of  repre- 
senting as  far  as  possible  the  latter  motion. 


*  It  will  be  uoticed  that  in  formiug  these  equatious  I  have  neither  used 
the  tiual  values  of  the  absolute  terms,  nor  taken  account  of  the  fact  that 
the  observed  motions  of  the  planes  are  referred  to  the  ecliptic.  Changes 
thus  produced  in  the  equations  are  too  minute  toaftect  the  conclusion. 


57,  58] 


ZODIACAL  LIGHT. 


115 


There  would  of  course  be  uo  dyuaraieal  impossibility  in  the 
hypothesis  of  a  single  planet  having  as  great  an  inclination  as 
that  required.  But  I  conceive  that  a  planet  of  the  adequate 
mass  could  not  have  remained  so  long  undiscovered.  Whether 
we  regard  the  matter  as  a  planet  or  a  ring,  a  simple  computa- 
tion shows  that  its  mass,  if  at  the  Sun's  surface,  would  be 

about  j-TT^  that  of  the  Sun  itself,  and  one-fourth  of  this  if  at  a 

distance  etpial  to  the  Sun's  radius.  We  may  conceive,  if  we 
can  not  compute,  how  much  light  such  a  mass  of  matter  would 
reflect.  Altogether,  it  seems  to  me  that  the  hypothesis  is 
untenable. 


ler  used 
ct  that 
hanges 
iou. 


Hypothesis  of  an  extended  mass  of  diffused  matter  like  that  which 
reflects  the  zodiacal  light, 

58.  The  phenomenon  of  the  zodiacal  light  seems  to  show 
that  our  Sun  is  surrounded  by  a  lens  of  diffused  iuatter  which 
extends  out  to,  or  a  little  beyond,  the  orbit  of  the  Earth,  the 
density  of  which  diminishes  very  rapidly  as  wo  recede  from 
the  Sun,  The  question  arises  whether  the  total  mass  of  this 
matter  m.ay  not  be  sufficient  to  cause  the  observed  motion. 

So  far  as  the  action  of  that  portion  of  matter  which  is  near 
the  Sun  is  concerned,  the  conclusions  just  reached  respecting 
a  ring  surrounding  the  Sun  will  apply  unchanged,  because  we 
may  regard  such  a  mass  as  made  up  of  rings.  Observation 
seems  to  show  that  the  lens  in  (piestion  is  not  much  inclined 
to  the  ecliptic,  and  if  so  it  would  produce  a  motion  of  the 
nodes  of  Venus  and  Mercury  the  opposite  of  that  indicated 
by  the  observations. 

There  is  another  serious  diffimilty  in  the  way  of  the  hypoth- 
esis. A  direct  motion  of  the  perihelion  of  a  planet  may  be 
taken  as  indicating  the  fact  that  the  increase  of  its  gravitation 
toward  the  Sun  as  it  passes  from  aphelion  to  perihelion  is 
sUghtly  greater  than  that  given  by  the  law  of  the  inverse 
square.  This  in(!rease  would  be  produced  by  a  ring  of  matter 
either  wholly  without  or  wholly  within  the  orbit.  But  if  we 
suppose  that  the  orbit  actually  lies  in  the  matter  composing 
such  a  ring,  the  effect  is  the  opposite;  gravitation  toward  the 


h 
it" 

r 

1  i 

1i 
1 1 

!  I 


110  EXTRA.MEBCUIIIAL  QROUP.  f.j8,  59,  GO 

Sun  is  relatively  diminished  as  the  planet  passes  from  aphelion 
to  perihelion,  and  the  motion  of  tuo  perihelion  would  bo  retro- 
grade. 

It  can  not  be  supposed  that  that  part  of  the  zodiacal  light 
more  distant  from  the  Sun  than  the  aphelion  of  Mercury  is 
even  as  dense  as  thai  portion  contained  between  the  aphelion 
and  the  i)erihelion  distantes.  The  result  in  <iuestion  must 
therefore  be  due  wholly  to  that  part  of  the  matter  which  lies 
near  to  the  Sun,  and  we  thus  have  all  the  ditTicultiea  of  the 
intra-mercurial  ring  theory,  with  one  more  added. 

Hypotheais  of  a  ring  (//  pit  me  to  id  a  hetwceti  the  orbits  of  Mercury 

and  Venus. 

.■)9.  It  appears  that  any  ring  or  zone  of  matter  adequate 
to  produce  the  observed  eft'ect  must  lie  between  the  orbits  of 
^lercury  and  Venus.  Its  assignment  to  this  position  requires 
a  UKU'e  careful  determination  of  its  possible  eccentricity. 
There  will  be  six  independent  elements  to  be  determined; 
the  mass,  the  mean  distance,  the  eccentricity,  the  perihelion, 
the  inclination,  and  the  node. 

I  find  that  the  observed  excesses  of  motion  of  the  elements 
of  ^Mercury  and  Venus  will  be  approximately  represented  by 
elements  not  dittering  much  from  the  following: 

Total  mass  of  group.     ......       37  (jj^,^,^^^ 

Mean  distance .  0.48 

Eccentricity  of  orbit 0.04 

Longitude  of  perihelion      .     .    .    .    .  10° 

Longitude  of  node 35° 

Inclination  to  ecliptic* 7^.5 

Probable  diameter  at  distance  unity  if 

agglomerated  into  a  single  planet    .  .'3".5 

Considerations  on  the  admissibility  of  the  hypothesis — Possible 
mass  of  the  minor  planets. 

60.  Although  the  preceding  hypothesis  is  that  which  best 
represents  the  observations  of  Mercury  and  Venus,  we  can 
not,  in  the  present  condition  of  knowledge,  regard  it  as  more 
than  a  curiosity.    True,  it  i&  plausible  at  first  sight.     Since^ 


GOI 


POSSIBLE    ACTION   OF   THE   MINOU    PLANETS. 


117 


as  already  remarked,  any  disturbing  body  of  sutticient  mass 
to  cause  the  observed  excess  of  motion  of  the  ]>erihelion  of 
Mercury  would  change  the  jiosition  of  the  planes  of  the  orbitfs, 
and  since  observations  give  a])i)arent  indications  of  such  a 
<'hauge  in  the  i)lane  of  the  orbit  of  Venus,  it  might  appear 
that  we  have  here  a  very  good  ground  for  tlie  view  that  all 
the  motions  are  due  to  the  attraction  of  unknown  masses. 
But  the  great  diniculty  is  that  the  excess  of  motion  of  the 
orbital  planes  is  in  the  opposite  direction  from  what  we  should 
expect.  A  grou])  of  bodies  revolving  near  the  phine  of  the 
ecliptic  would  jtroduce  a  retrograd<^  motion  of  the  nodes.  But 
the  observed  excess  is  direct.  A  <lirect  motion  can  be  pro- 
duced only  in  case  the  orbits  are  more  inclined  than  those  of 
the  disturbed  planet.  In  admitting  such  orbits  we  encounter 
dilliculties  which,  if  not  absolutely  insurmountable,  yet  tell 
against  tlie  vnobability  of  the  hyi»(»thesi8. 

The  hypothesis  (tarries  with  it  the  probable  result  that  the 
excess  of  motion  of  the  i)erihelion  of  Mars  is  produced  by  the 
action  of  the  minor  jdanets.  1  have  considered  the  (|uestion 
of  this  action  in  an  unpublished  investigation.  From  the  i)rob- 
able  albedo  and  magnitudeof  the  minor  planets  and  the  obser 
vatious  of  JiARNAK'D  and  others  on  their  diameters,  1  liave 
determined  the  probable  massofeach  partof  the  grou])  having 
a  given  opposition  magnitude.  The  result  is  that  the  number 
of  these  bodies  having  such  a  unignitude  appears  to  progress 
in  a  fairly  uniform  nuinner  through  several  nnignitudes.  The 
ratio  of  progression  may  lie  anywhere  between  tlie  limits  2 
and  3.  Up  to  the  limit  .'{  the  total  mass,  if  continued  on  to 
inhnity,  could  not  produce  any  appreciable  ett'ecton  the  motion 
of  ]\I«rs.  But  if  we  suppose  a  larger  ratio  than  3  to  prevail, 
then  the  number  of  planets  of  smaller  magnitude  would  be  so 
numenms  as  to  form  a  zoneof  liglrt  across  the  heavens,  as  may 
readily  be  seen  by  considering  that  the  total  amount  of  light 
retlected  from  the  planets  of  each  order  of  magnitude  would 
form  an  increasing  series,  since  the  ratio  between  the  brillian- 
cies of  two  objects  of  unit  difference  in  magnitude  is  only 
about  2.5.  We  umy  therefore  suppose  that  the  faint  band  of 
light  which  is  said  to  be  visible  across  the  entire  heavens  as 
a  continuation  of  the  zodiacal  light,  as  well  as  the  "gegeu- 


118 


hall's  hypothesis. 


f(K),  «1 


scliein,"  is  due  to  these  ininute  bodies,  and  yet  find  tlieir  total 
mass  too  small  to  i>rodnce  any  ai)i)ie('iable  etlect. 

Whether  we,  can  assign  to  the  components  of  such  a  group 
any  magnitude  so  small  that  they  would  be  individually  invis- 
ible, aud  a  number  so  small  that  they  would  not  be  seen 
collectively  as  a  band  of  light  brighter  than  the  zodiacal  arch, 
and  yet  having  a  total  mass  so  large  as  to  produce  the  observed 
eflects,  is  a  very  imp(U'tant  (piestion  which  can  not  be  decided 
without  exa<!t  photometric  investigations.  It  is,  however,  cer- 
tain that  if  we  could  do  so  we  should  have  to  suppose  a.  very 
unlikely  discontinuity  in  the  law  of  ])rogression  between  each 
magnitude  and  the  numbt'r  of  bodies  having  that  magnitude. 
It  must  therefore  suHice  for  our  present  object  that  we  regard 
the  hypothesis  of  such  bodies  as  unsatisfactory. 


i 


Hypothesis  that  (fntvitatioti  toirar<(  the  sun  is  not  exactly  as  the 
inverse  square  of  the  distance. 

(>1.  Prof.  Hall's  hypothesis  seems  to  me  provisionally  not 
inadmissible.  It  is,  that  in  the  expression  for  the  gravitation 
between  two  bodies  of  masses  m  and  m'  at  distance  /• 


Force  =  "JUL 


the  exponent  n  of  r  is  not  exactly  2,  but  2  +  rf,  d  being  a  very 
small  fraction.  This  hypothesis  seems  to  me  much  more 
simple  and  unobjectionable  than  those  which  suppose  the 
force  to  be  a  more  or  less  complicated  function  of  the  relative 
velocity  of  the  bodies.  On  this  hypothesis  the  perihelion  of 
each  planet  will  have  a  direct  motion  found  by  multiplying  its 
mean  motion  by  one-half  the  excess  of  the  exponent  of  grav- 
itation. 


Putting 


»  =  2.000  000  1574 


the  excess  of  motion  of  each  i>erihelion  of  the  four  inner 
planets  would  be  as  follows.  It  will  be  seen  that  the  evidence 
in  the  case  of  Venus  and  the  Earth  is  negative,  owing  to  the 


oil 


LAW  OP  GRAVITATION. 


119 


very  siiiull  eccentruities  of  their  orbitvS,  while  tlie  observed 
luotiou  iu  the  ease  of  Mars  is  very  eh)sely  re|>resente(l. 


Dt^r 

«I)t;r 

// 

// 

Merniry, 

42..^4 

8.70 

Venus, 

l^.-'iS 

0.11 

Kartli, 

lO.L'O 

0.17 

Mars, 

oA2 

0.51 

All  iiidepeinlent  test  of  this  hyi)othesis  in  the  case  of  otiier 
bodies  is  very  desirable.  The  only  case  in  which  there  is  any 
hope  of  deterniininji:  such  an  excess  is  that  of  the  Moon,  where 
the  excess  wouhl  amount  to  about  140"  per  century.  This  is 
very  nearly  the  hundred-thousandth  i)art  of  the  total  motion 
of  the  perigee.  The  theoretical  motion  has  not  yet  been  com- 
puted with  (juite  this  degree  of  precision.  The  only  determi- 
nation which  aims  at  it  is  that  made  by  Hansen.*     lie  finds 


Theory.  01)8cr.  Diff. 

//  //  // 

Annual  mot.  of  perigee,     140  434.04;  IMM.i.l.OO;  4-1.5(3 

Annual  mot.  of  node,        -00  070.70  -0(m;70.02;  -L'.SO 

The  observed  excess  of  motion  agrees  well  with  the  hy]»oth- 
esis,  but  loses  all  sustaining  force  from  the  disagreement  in 
the  case  of  the  node.  The  difterences  Hansen  attributes 
(wrongly,  I  think)  to  the  deviation  of  the  figure  of  the  Moon 
from  mechanical  sphericity. 

Conftifttenvji  of  HaWs  hypothesis  with  the  ff€)i€ral  results  of  the 

law  of  gravitation. 

02.  The  law  of  the  inverse  scjuare  is  proven  to  a  high  degree 
of  approximation  through  a  wide  range  of  distances.  The  close 
agreement  between  the  observed  parallax  of  the  ^loon  and 
that  derived  from  the  force  of  gravitation  on  the  Earth's  sur- 
face shows  that  between  two  distances,  one  the  radius  of  the 
Earth  and  the  other  the  distance  of  the  Moon,  the  deviation 
from  the  law  of  tlie  square  can  be  only  a  small  fraetion  of  the 


* Darlegung,  etc.:  Abhandhungen  der  Math.-Phys.  Cluase  der  lion.  Saehai- 
Bchen  (ieselhchaft  der  Wisienvchaften ,  vi,  p.  348. 


-^«-  ».ju*  '■ 


lao 


hall's  IIYPOTIIESIH. 


(«2 


thousaiKltli  jmit,  <»r,  wo  iiuiy  Huy,  u  quantity  of  th«^  or(l(>r  of 
maKiiitixU' of  tli(>  livc-tlioiiHuiMllli  ))art. 

Coiiiiiij^  down  to  snialU'i-  distaniM's,  w«  llnd  that  the  <;lo8e 
ajjieemcnt  h'-twocn  the  dt-naity  of  the  Earth  as  derived  troni 
the  attraction  of  small  masses,  at  distances  of  a  fraction  of  a 
meter,  with  th«^  density  whi«h  we  niijjht  /<  priori  snpposo  the 
Earth  to  hav*-,  shows  that  within  a  ranjje  of  distance  extend- 
ing from  less  than  one  meter  to  more  than  six  million  meters, 
the  acenmiilated  deviation  from  the  law  can  scarcely  amount 
to  its  third  part.  The  coincidence  of  the  disturbin}^  force  of 
the  Snn  ui)on  the  Moon  with  that  computed  upon  the  theory 
of  jjravitation,  extends  the  coincidence  from  the  distance  of 
the  Moon  to  that  of  the  Snn,  while  Ivhplkr's  third  law 
extends  it  t(>  the  outer  planets  of  the  system.  Here,  however, 
the  result  of  observations  so  far  nuide  is  relatively  less  pre- 
cise. We  may  therefore  say,  with  entire  coulidence,  as  a 
result  of  accurate  measurement,  that  the  law  of  the  inverse 
square  holds  true  within  its  live-thousandth  part  from  a  dis- 
tance e(|ual  to  the  I^arth's  radius  to  the  distance  of  the  Sun,  a 
rauf^e  of  twenty  lour  thousand  times;  that  it  holds  true  within 
a  third  of  its  whole  amount  throu;;h  the  ranye  of  six  million 
times  from  one  meter  to  the  Earth's  radius;  and  within  a 
small  but  not  yet  well-defined  quantity  from  the  distance  of 
the  Sun  to  that  of  Franus,  in  which  the  multiplicatiou  is 
tweutyfold. 

J  ni  all's  hypothesis  contradicted  these  couclusious  it  would 
be  untenable,  liut  a  very  simple  computation  will  show  that, 
assuming*'  the  force  to  vary  as  r-  -  '  * ,  6  beinj^  a  minute  cou- 
stant  sullicieut  to  acccmnt  for  the  motion  of  the  perihelion  of 
INIercury,  the  effect  would  be  entirelj'  inappreciable  in  the  ratio 
of  the  gravitation  of  any  two  bodies  at  the  widest  range  of 
distance  to  which  observation  has  yet  extended.  Although 
the  total  action  of  a  mate'ial  point  on  a  si)herical  surface  sur- 
rounding it  would  converge  to  zero  when  the  radius  became 
infinite,  instead  of  remaining  constant,  as  in  the  case  of  the 
inverse  square,  yet  the  diminution  in  the  action  upon  a  surface 
no  larger  than  would  suiiHce  to  include  the  visible  universe 
would  be  very  small. 


■/ 


m\ 


roilKKl  TION   OF   MASSES. 


121 


MasNrs  it/ the  planets  irliivli   rrpt'i'Hviit  the  neoular  nd'hitionn  of 
other  eh'inentx  thmt  the  perihelia. 

<I3.  Oil  IIam/8  liypotlit'His  tlu'  siMiilar  vsiriiitioiis  »»r  all  tlio 
clciiiciits  otlici  than  the  p('i'ili(>1ia  will  rciiiaiii  iiiicliaii^«Ml. 

Oiir  iM'xt  i)r(»l)U'm  is  to  coiisidrr  the  possibility  of  leincsont- 
iiij^-  tin'  \ariatioiis  of  tln^  otluT  t'h'iiH'iits  l»y  admissil)!)'  masses 
ol"  tln'  known  plaiu'ts.  in  i  .Vi  1  have  yiviMi  a  conipaiisoii  of 
tlie  scculai'  variations  as  tlu-y  result  from  obsiMvafions.  with 
tlu'ir  tht'orotical  exjni'ssiims  in  terms  of  corrections  to  a  cer 
tain  system  of  masses.  When  the  e(iiuiti«»ns  thus  formed  are 
multiplied  by  the  fa<'tors  V ir.  Mhieh  make  the  mean  erroi- of 
ea«'h  e(|uatifMi  unity,  we  have  the  followinj^'  system  of  equa- 
turns,  in  which  we  jnit  /'  =  lO.r; 

O.r     +     {', ,.'  4-     1'  ,-"     +  (»  ,' 

0—1  —      L'                   (» 

_   7        _i(),s  -  27          -  .". 

0  -  L'  1             -    1 

0  0-1 

-L'.ii       -;u(;        -i(( 
-f  i;{  0        -1(5 

-iL'U  0        -  ;i 

0-1-8  0 


-0.) 

+  2.-i 

+  -'1 
-lli 
—   0 

+   I 


-11 


+   (50 
-12(5 


0 


=  -  1.7 

=  -I-  o.r» 

=  4-  O.a 
=  4-  0.0 

==  +  i.:> 

=  4  4.2 
=  4-  0.1' 
=  -  2.0 

r=.    +   1.1 

=  -I-  0.-1 

=  -  0.8 


-  1.8 

+  0..-. 

-t-1.1 
4-0.7 
+  l.a 
0.0 
+  0.1 
-0.7 

+  i.;j 

-0.2 
-0.2 


The  resulting;  noruuil  ecpiations  are 

.-,70(;,,.  _  ir.(5;i;''-  40!>1  /'"  4-  1 10  i'"'  =  -(-  lU 
-  15G;{  +  101231  4-  88r)5(5  -|-  .34.~m  =  -  <570 
_  4901      4.    88.M(5        -I-  122 1(52       -|-  IMrA)         =  -  1 14(5 

4.   140    -f     .'54r)r)      4-     ;}7r)0      4-    101       =  _    ;',<i 

Along-  with  the  results  of  the  solution  (»f  these  etpKitions  I 
place,  for  comparisou,  the  xalues  of  Chu])ter  V,  which  have 
been  considered  most  ]»robable. 


From  HOC.  v.ir. 


10.1=  r      =-f  0.070 


From  other  sources. 
4-  0.08        i  0.20 


v'  =  -I-  0.0100  ±  .0050  +  0.0084  ±  0.0028 
r"  =  -  0.0183  ±  .0052  -  0.00304  i  0.0015 
r'"  =  -  0.0115         ±  .0(57  +  .0037      i  0.018 


i 


11  r 


'4 


!  1: 


lU 


m 


til 


122 


CORRECTION  OF  MASSES. 


[63,64 


By  substitution  in  the  couditioual  equations  we  find  for  the 
mean  error  corresponding  to  weight  unity— 

fi  =  i  1.14 

In  forming  these  equations  they  were  reduced  by  multipli- 
cation to  a  supposed  mean  error  of  L  1.  Speaking  in  a 
general  way  we  may  therefore  say  that  the  representation  of 
the  secul.ar  variations,  those  of  the  perihelia  being  ignored, 
by  these  corrections  to  the  masses  is  satisfactory.  Except  for 
the  large  discordance  in  the  motion  of  the  eccentricity  of 
Mercury  the  mean  error  would  have  been  less  than  unity. 

Comi)aring  the  two  sets  of  values  we  find  that  the  masses 
of  Mercury,  Venus,  and  Mars  agree  well  with  those  derived 
from  other  sources.  Very  ditterent  is  it  with  the  mass  of  the 
Earth.  The  discordance  ivS  here  more  than  the  hundredth 
part  of  its  whole  amount,  whidi  involves  a  discordance  of 
more  than  tlio  three  hundredth  part  in  the  value  of  the  solar 
parallax.  Let  us  now  proceed  in  the  reverse  order,  and  deter- 
mine the  value  of  the  solar  pan'!lax  from  the  mass  of  the  Earth, 
as  derived  from  the  preceding  data. 

Preliminary  (uljusimvut  of  the  two  .sets  of  masses. 

04.  We  nmke  the  best  adjustment  for  this  purfose  by  adding 
to  the  eipiations  of  conditiim  last  given  the  additional  ones 
derived  from  the  values  of  the  masses  discussed  in  Chapter  V. 
Multii)lyin;;'  each  vahnr  of  r  by  the  fiictor  necessary  to  reduce 
the  mean  error  of  the  second  member  of  the  eiiuation  to  unity, 
we  have  the  following  conditi  nal  equations: 


50  ,v     =  -f  0.4 
300  y'    =  -f  2.9 
50  i'"'  =       0.0 
30  y'"  =  -f  0.42 

Of  the  last  two  equations  it  may  be  remarked  that  the  first  is 
that  given  by  Prof.  Hall's  original  mass  of  1877,  while  the 
last  is  derived  by  Dr.  IIarshman  from  Hall's  observations 
of  the  outer  satellite  made  during  the  opposition  of  1892. 


64J 


CORRECTION  OF  MASSES. 


123 


When  we  add  to  the  normal  equations  already  formed  the 
products  of  these  last  equations  by  the  factors  of  the  unknown 
quantities,  the  system  of  normal  equations  is  as  follows : 


82r)<>j' 

-     1563  v' 

-     4091  v" 

+   140j'"' 

=  +134 

-1503 

+  230831 

4-  88556 

+3455 

=  +374 

-49!H 

+  88556 

+  122462 

+3750 

=  -1446 

+  140 

+     3455 

+     3750 

-i-  3S01 

=  -26 

The  solution  oftliese  equations  ftives  the  following  values  of 
the  unknown  quantities: 

.1-    =  +  0.0071  i:  .0120 
/'    =  +  0.071    +  .120 
y'   =  +  0.0084  i  .0024 
/  "  =  -  0.0177  ±  .0035 


u"'  — 


+  0.0027 


.01(5 


Here  aj^ain  we  note  that,  the  Earth  aside,  the  results  for  the 
masses  are  quite  satisfactory.  The  correction  to  Prof.  Hall's 
original  mass  of  ^lars  is  so  minute  and  so  much  less  than  its 
probahki  error  that  we  may  consider  this  value  of  the  mass  to 
be  confirmed,  and  may  adopt  it  !•■>>  Jefinitive  without  question. 
The  corrections  to  the  masses  of  Mercury  and  Venus  are  scarcely 
changed.    The  mean  residu.il  is  reduced  to 

f  =  i  0.01 

which  is  less  than  the  supposed  value. 

We  have,  therefore,  so  far  as  these  results  go,  no  reason  for 
distrusting  the  following  value  of  the  solar  parallax,  which 
results  from  that  of  the  mass  of  the  Kanh  thus  derived: 

7T  =  8".750  L  ".010 

Tiie  critical  cvamination  and  comjiarison  of  this  and  other 
values  of  the  i)arallax  is  the  work  of  the  next  two  chapters. 


r 


ill 


II;" 


if ' 


if 


Cn AFTER  VII. 

VALUES  OK  THE  PRINCIPAL  CONSTANTS  WHICH  DEFINE 
THE  MOTIONS  OF  THE  EARTH. 

The  Pnvensional  Constant. 

05;  The  accurate  dctcriiiiiuitioii  of  the  annual  or  centennial 
motion  of  precession  is  somewhat  (lillicult,  owing  to  its  depend- 
ence on  several  distinct  dements,  and  to  the  probable  system- 
atic errors  of  the  older  observations  in  Right  Ascension  and 
Decimation.  What  is  wanted  is  the  annual  motion  of  the 
e(|uinox,  arising  from  the  combined  motions  of  the  eijuator 
and  ilu!  eclii>tic,  relative  to  <lirections  absolutely  fixed  in  space. 
As  observations  can  not  be  referred  to  any  line  or  plane  which 
we  know  to  be  ji  bsoluteiy  H.ved,  we  are  obliged  to  assume  tlia  t  ♦^he 
general  Tuean  direction  of  tJM^  fixed  stars  remains  uncb  r.^vw 
or.  ill  other  words,  that  the  stellar  system  in  general  has  no 
motion  of  rotation.  Tliis  is  a  wife  assuini)tion  "-'o  far  as  the 
great  mass  of  stars  of  smaller  magnitude  is  cone*  rned.  Bur  it 
is  not  on  such  stars  that  we  have  the  earliest  accurate  obser- 
vations. Moreover,  observed  Right  Ascensions  of  these 
fainter  stars  rt'iative  to  the  brigliter  ones  are  subject  to  i)ossi- 
ble  systematic  errors,  arising  from  thei)ersonal  equation  being 
ditferent  for  brighter  iind  fainter  stars.  In  the  case  of  the 
stars  observed  by  1>|{M)Lhv,  there  is  frequently  such  commu- 
nity of  proi»er  motion  among  neighboring  stars  that  we  <.'an 
not  be  (juite  sjire  that  all  rotation  is  eliminated  in  the  general 
mean.  Tuder  these  circumstances  we  liave  only  to  make  the 
best  use  that  we  can  of  existing  material. 

We  must  also  reuu'inber  (hat  observed  Itight  Ascensions  are 
not  directly  referred  to  the  efjuinox,  but  to  the  ►Sun,  of  which 
the  error  of  absolute  nu'an  Right  Ascension  must  bo  det*^^ 
mined.  This  again  can  be  done  only  from  observed  declina- 
tions, since  by  delinition  the  equinox  is  the  point  at  which 
the  Sun  crosses  the  e(iuator.  It  is  also  to  be  noted  that  the 
clock  stars  which  are  directly  compared  with  the  Sun  by  no 
124 


65] 


THE  PRECESSIONAL  CONSTANT. 


125 


the 


means  include  the  whole  list  to  l)e  used  aa  absolute  points  of 
reference.  We  therctbre  have  three  separate  steps  in  (letenjiin- 
iug  completely  a  correction  to  the  adopted  annual  precession: 

(1)  The  correction  to  tlie  Sun's  absolute  mean  Hij;ht  Ascen- 
sion or  longitude. 

(2)  The  correction  to  the  general  mean  Ifight  Ascension  of 
the  clock  stars  relative  to  the  Sun. 

(3)  The  determination  of  the  clock  stars  relative  to  the  great 
mass  of  stars. 

It  goes  without  saying  that  the  determinations  of  these  three 
quantities  are  entirely  indepen<lent  of  each  othci-,  and  that  the 
])recision  of  the  result  depends  on  the  precision  of  each  sepa- 
rate determination. 

The  motion  of  the  ])ole  of  the  e(|uator,  on  which  the  luni- 
solar  precession  depends,  may  W,  determined  by  observed 
Declinations  ([uite  iudepemlently  of  the  Right  Ascensions.  A 
determination  of  the  precession  from  the  latter  includes  tl»e 
planetary  prec':s8ion,  but  as  this  has  to  be  determined  from 
theory  independently  of  observations,  we  have,  in  observed 
Kight  Ascensions  and  Decilinations,  two  independent  methods 
of  determining  the  motion  of  the  equator. 

It  fortunately  happens  that  the  constant  of  i)recession  is 
not  so  closely  connected  with  other  constants  that  a  small 
error  in  its  determination  will  seriously  affect  our  geueral  con- 
clusions, or  the  reduction  of  places  of  the  fixed  stais,  because 
the  eft'ect  of  an  error  will  be  nearly  eliminated  through  the 
proper  motions  of  the  fixed  stars,  or  the  motums  of  the  planets 
in  longitude.  I  have  therefore  satisfied  myself  with  reviewing 
and  combining  the  four  best  determinations. 

1  pass  over  in  silence  the  classic  determinations  of  Bessel 
and  Otto  Struve,  because  the  material  on  which  they  depend 
has  been  incorporated  in  more  recent  works.  Of  these  the  one 
which  seems  entitled  to  most  weight  is  that  of  LuDWUrSTRUVK, 
Bestimnntnfi  (hr  Con,stante  der  Prfwessioti,  und  der  eujenen 
Bewegung  des  Sonnemystemn.*  Tiiis  work  was  suggested  by 
the  completion  of  Auwers'  re-reduction  of  Bradley's  Obser- 
vations, and  of  the  Pulkowa  standard  catalogues  for  184'), 

*M6inoire8  de  I'Acndi^inie  Impdrialo  des  Scieocea  do  St.  Pdtersbourg. 
VII"^^  8<5rie.    .Tome  xxxv,  No.  3. 


■  I 


yii 


f: 


in 


t  * 


il 


120  THE  PRECESSIONAL  CONSTANT.  [65 

185o,  and  1805.  It  depends  entirely  on  the  Bradley  stars, 
and. the  result,  when  reduced  to  the  most  probable  equinox, 
may  be  regjirded  as  the  best  now  derivable  from  those  stars, 
or,  at  least,  as  not  susceptible  of  any  large  correction. 

He,  of  course,  includes  in  his  work  the  determination  of  the 
motion  of  the  solar  system  relative  to  the  mass  of  the  stars. 
In  addition  to  this,  the  possibility  of  a  common  rotation  of 
the  Bradley  stars  around  the  axis  of  the  Milky  Way  is  con 
sidered.  This  rotation  I  should  be  disposed  to  regard  as  zero 
for  the  present. 

In  place  of  considering  each  of  the  2,509  stars  singly,  he 
divides  the  celestial  sphere  into  120  spherical  trapezoids,  each 
covering  15  degrees  in  Declination,  and  an  arc  of  laght 
Ascension  equal  approximately  to  one  liour  of  a  great  circle 
at  the  equator.  The  questicui  might  be  legitimately  raised 
whetl  •  ^  different  system  of  Aveighting  the  trapezoids,  founded 
on  a  coi.  vtion  and  comparison  of  the  projjer  motions  in 

Right  Asci  i. oion  and  Declination  would  not  have  been  advis- 
able. I  am,  however,  fairly  confident  that  no  change  in  this 
respect  would  have  materially  affected  the  result.  With  this 
work  of  Strive  1  have  combined  those  of  Bolte,  Dreyer, 
and  ]S^VRKN. 

In  the  case  of  the  Right  Ascensions  it  is  necessary  to  reduce 
all  the  results  to  the  equinox  determined  in  the  last  chapter. 
From  this  chapter  it  appears  that  the  standard  Right  Ascen- 
sions with  which  the  redm^tion  of  the  preceding  investigations 
have  been  made  require  a  (correction  to  the  centennial  motion 
of  -f  0".3(>.  Reducing  each  determination  to  the  equinox  thus 
defined,  we  iuive  the  following  results  for  the  general  preces- 
sion in  Right  Ascension  at  the  epoch  1800: 
L.  Stritve,  from   the  comparison  of 

Auwers  Bradley  with  the  modern 

I'ulkowa  Right  Ascensions      .     .     .  mj  =  40".050l ;  ?r  =  4 
Drevkr,    from    the    comparison    of 

LaLande's  Right  Ascensions  with 

those  of  Schiellerfp 40  .0011;  w  =  2 

Nyrkn,  by  the  comparison  of  BesselV, 

Right    Ascensions    with     those    of 

Schjellerup 40  .0450;  w  =  l 

Mean 40  .0520 


[65 


65]  THE  PREOESSIONAL  CONSTANT.  127 

The  weights  here  assigned  are  of  course  a  matter  of  judgment. 
Tlie  general  agreement  of  the  results  is  as  good  as  we  could 
expect. 
From  observed  declinations  we  have — 

L.  Struve,   from  the  comi)arison  of 

Atwers  -  Bradley    with    modern 

Pulkowa  catalogues «  =  20".0495;  »r  =  2 

BoLTE,  from  the  comparison  of  La- 

lande's  Declinations  with  those  of 

Scnj.ELLERUP 20  .0537 ;  w  =  1 

Mean 20  .0500 

We  have  now  to  combine  these  independent  results.  I  ]>ro- 
pose  to  call  Precessional  Constant  that  function  of  the  masses 
of  the  Sun,  Earth,  and  Moon,  and  of  the  elements  of  the  orbits 
of  the  Earth  and  Moon,  which,  being  multiplied  by  half  the 
sine  of  twice  the  obliquity,  will  give  the  annual  or  centennial 
motion  of  the  pole  on  a  great  -circle,  and  being  nndtii)lied  by 
the  cosine  of  the  obliquity  will  give  the  lunisolar  precession 
at  any  time.  It  is  true  that  this  quantity  is  not  absolutely 
constant,  since  it  will  change  in  the  course  of  time,  through 
the  diminution  of  the  Earth's  eccentricity.  This  change  is, 
however,  so  slight  that  it  can  become  appreciable  only  after 
several  centuries.    If,  then,  we  put 

p,  the  precessional  constant,  we  have,  for  the  annual  general 
precession  in  Right  Ascension  and  Declination — 

m  =  P  cos'^  e  —  «  sin  L  cosec  6 
M  =  P  sin  t  cos  5 

L  being  the  longitude  of  the  instantaneous  axis  of  rotation 
of  the  ecliptic,  and  u  its  annual  or  centennial  motion.  From 
the  definitive  obliipiity  and  masses  of  the  planets  ad(q>ted 
hereafter,  we  find  the  following  values  of  «,  L,  and  e,  for  1800 
and  1850: 

1800.  1H50. 

.       log«=        1.07372;  1.07341 

L  =  1730    2'.31 ;  1730  29'.08 

f=    23    27.92;  23    27.53 


128 


THE  PRECESSIONAL  CONSTANT. 


We  thus  ttiul  the  following  values  of  r,  the  unit  of  time 
being  100  solar  yeais: 


From  IkigUt  Ascensions, 
From  Declinations, 


p  =  r)490.12;  w  =2 
r  =  5489.44;  n'  =  l 


Mean,  P  =  5480".89 

As  the  data  used  in  Strttve's  investigation  may  be  con- 
sidered of  a  more  certain  kind  than  those  used  by  the  others, 
we  may  compare  these  results  with  those  which  follow  from 
Stbuve's  work  alone.     They  are 


// 


From  liight  Ascensions, 
Fi'oni  Declinations, 


P  =  5489.83 
P  =  5489.00 


Giving  double  weight  to  the  results  from  the  Right  Asceu 
sion^>.,  the  results  may  be  expressed  as  follows : 


// 


From  Struve's  investigation,    p  =  .">489.57 
From  the  other  two  works,  P  =  5490.18 

B«'fore  concluding  this  investigation,  I  had  adopted  as  a  pre- 
liminary value 

P  =  5489".78 

As  this  result  does  not  differ  from  the  one  I  consider  most 
probable,  548!>".89,  by  more  than'  the  probable  error  of  the 
latter,  and  diverges  fiom  it  in  the  direction  of  the  best  deter- 
mination, I  have  decided  to  adhere  to  it  as  the  detinitive 
value. 

The  centennial  value  of  P  is  subjected  to  a  secular  diminu- 
tion of  0".00;?64  per  ceutnry,  owing  to  the  secular  diminution  of 
the  eccentricity  of  the  Earth's  orbit.    We  therefore  adopt 

//  // 

p  =  5489.78  -  0.00304  T  for  a  tropical  century. 
p  =  5489.90  -  0.00304  T  for  a  Julian  century. 

In  the  use  of  P  I  at  first  neglected  the  secular  variation, 
but  have  added  its  efiect  to  the  results  developed  in  powers 
of  the  time. 


66]  THE  CONSTANT  OF  NVTATION. 

Constant  of  nntai'm  ilerivid  from  obscrrathnts. 


129 


<;6.  The  determination  of  this  constant  from  observationH  is 
extremely  satisfactorj',  owing  to  tlie  c<miph'teness  with  which 
systematic  errors  may  Im'  eliminated.  If,  with  a  meridian 
instrument,  reynlar  observations  are  made  through  a  draconitic 
perio<l,  on  a  uniform  plan,  upon  stars  ecpially  distributed 
through  the  circle  of  Itight  Ascension,  the  observations  being 
made  daily  through  more  than  1-  hours  of  Right  Ascension, 
all  systematit!  errors  in  the  determination  of  the  nadir  point 
and  all  having  a  <liurnal  or  annual  period  may  be  completely 
eliminated  from  the  constant  in  question.  These  conditions 
are  so  nearly  fullilled  in  the  observations  with  the  Greenwich 
transit  circle,  and,  to  a  less  extent,  in  those  with  the  Wash- 
ington transit  circle,  that  the  results  of  the  woric  with  those 
two  instruments  alone  are  entitled  to  greater  weight  than  has 
hitherto  been  sn[»posed.  I  have,  however,  discussed  quite 
fully  all  previous  determinations  of  which  it  seemed  that  the 
probable  mean  error  would  be  less  than  J-  ()".!(). 

Keferring  to  the  volume  on  the  subject  to  be  hereafter  pub- 
lished, the  results  of  the  discussion  are  presented  in  the  fol- 
lowing table.  The  weights  are  assigned  ou  the  supposition 
that  weight  unity  shoidd  corres^wnd  to  a  mean  error  of  about 
±  0".07,  or  to  a  i)robable  error  of  i  0".05,  this  probable  value 
being  not  entirely  a  matter  of  comi>utation  from  the  discord- 
ance of.  the  sepfirate  results,  but,  to  a  certain  extent,  a  matter 
of  judgment. 

It.nurst  be  understood  that  the  results  below  are  not  always 
those  given  by  the  authors  who  are  (pioted,  but  that  their  dis- 
cussion has,  wherever  possible,  been  subjected  to  a  revision  by 
the  introduction  of  modern  data,  or  by  what  seemed  to  me 
improved  combinations.  Thus,  Nvren's  ecpmtions  have  been 
reconstructed  on  a  system  slightly  ditterent  from  his,  and  have 
been  corrected  forCiiANDLER's  variaticm  of  latitude.  Peteks's 
classical  work  has  also  been  corrected  by  the  introduction  of 
later  data,  and  by  a  resolution  of  his  equations.  The  Green- 
wich and  Washington  results  have  been  derived  from  the  dis- 
cussion in  Astronomical  Papers,  Vol.  II,  Part  VI. 
5600  N  ALM 9 


it! 


W.  ' 


1.50  THE  CONSTANT   OF  NUTATION.  [()(» 

Valncn  offhe  constant  of  nutation  derived  from  ohxerrationH. 

HuscH,  from  Bhadlky's  ol>.servati<)iis  with 

the  zenith  .sector ".».2.{L'  1 

EoniNSON,  from  lliveiiwi<;h  mnrjil  circles  .     .  9.22  1 

Peters,  from  Right  Ascensions  of  Polaris    .  0.214  4 

Li'NDAiiL,  fnrni  Declinsitions  of  Polaris     .     .  9.2.'}<)  X.'y 

Nyr6n,  from  ?»  Urs.  ^laj 9.2r»4  3 

"         ''     oDracouis 9.242  2.5 

»<         ''      /  Draconis 9.240  4 

DkBall,  from  Wagner's  Ri{?ht  Ascensions 

of  Polaris t>.ir.2  3 

De  IUll,   from  Wagner's  Declinations  of 

Polaris 9.213  3 

DeBall,  from  Wagner's  Rijjht  Ascensions 

ofrilfephei 9.2r>2  3 

DeIUll,  from  Wagner's  Declinations  of 

51  ('e]>hei 9.227  3 

DeBall,  from  Wagner's  Right  Ascensions 

offHTrs.  Min 9.208  3 

De  Bael,  from   Wagner's  Declinations  of 

rf  Urs.  Min 9.203  3 

Greenwich  XorthPolar  Distances  of  Sonth- 

ern  Stars,  Series  I 9.110  3 

Green\v'ich  North- Polar  Distances  of  South- 
ern Stars,  Series  II 9.201  3 

Greenwich  North-Pohir  Distances.of  North- 
ern Stars,  Series  I 9.204  4 

Greenwich  North- Polar  Distances  of  North- 
ern Stars,  Series  TI 9.223  4 

Washington  Transit  Circle,  southern  stars    .  9.217  0 

«  "  "       northern  stars    .  9.1 77  3 

Greenwich,  Right  Ascensions  of  Polaris   .     .  9.1."»3  2 

'<  Declinations  of  Polaris  ....  9.242  2 

"  Right  Ascensions  of  51  Ceilhei    .  9.135  2 

"  Declinations  of  51  Cephei    .     .     .  9.102  2 

"  Right  Ascensions  of  fJ  Urs.  Min.  9.147  2 

"  Declinations  of  6  l^rs.  Min.     .     .  9.235  2 

"  Right  Ascensions  of  A  Urs.  Min.  9.1C1  1 

*<  Declinations  of  A  Urs.  Miu.      .     .  9.339  1 

Mean 9.210;  w<.=72 


«(;,()7 


PRECESSION   AND  NUTATION. 


131 


The  iin'iin  error  eorreapondiufj:  to  weight  unity  wlieii  derived 
from  tlie  diHcordance  of  the  results  i.s  I  (>".(M!8.  wliile  the 
estimate  was  i  0".07().  We  luuy  therefore  put,  as  the  resulj 
of  observation — 


KelatioHs  between  the  constatits  of  pr  eves  ft  ion  anti  nutation,  and 
the  qxanfit'es  on  irhirh  they  depend. 

(17.  The  foriuuhe  of  precession  and  nutation  liave  been 
developed  by  Oppolzer  w'th  very  great  rigor  and  witli 
great  numerical  completeness  as  regards  the  elements  of  the 
Moon's  orbit,  in  tlie  first  volume  of  his  Buhnhestimmnng  der 
Komcten  and  Planeten,  second  edition,  Leipzig,  188L'.  What 
is  remarkable  about  this  work  is  that  it  constantly  takes 
account  of  the  possible  difference  between  the  Earth's  axis 
of  rotation  and  its  axis  of  figure,  a  distinction  which  has 
become  emphasized  by  Chandler's  dis«u)very  since  Oppol- 
zer wrote.  His  theory  however  fails  to  take  account  of  the 
change  in  the  i)eriod  of  the  Eulerian  nutation  produced  Ivy 
the  mobility  of  the  ocean  and  the  elasticity  of  the  Earth.  I'«ut 
this  effect  is  of  no  importance  in  the  present  discussion. 

From  Oppolzer's  developments,  1  have  <lerived  the  follow- 
ing expressions,  in  which  the  numerical  coeflicients  nray  be 
regarded  as  absolute  constants,  so  accurately  determined  tluit 
no  (piestion  of  their  errors  need  now  be  considered.  These 
results  have  been  derived  (piite  independently  of  the  similar 
ones  by  Mr.  Hill  in  the  Astronomieal  Journal,  Vol.  Xl,  which 
are  themselves  indbj)endent  of  Oppolzer's  Mork.  In  these 
fornuilie  we  have — 

I^,  the  constant  of  lunar  nutation  of  the  obli(|uity  of  the 
ecliptic,  as  defined  by  the  equation  Jf  =  X  cos  Q,  and 
expressed  iu  seconds  of  arc; 

P,  so  much  of  the  precession  of  the  e(|uinox  on  the  fixed 
ecliptic  of  the  date,  in  seconds  of  arc  and  iu  a  Julian 
yeai",  as  is  due  to  the  action  of  the  Moon; 

P',  so  nnich  of  the  same  i)rece88ion  as  is  due  to  the  action 
of  the  Sun. 


!     I 


V'' 

'1-  > 


I, 


182  MASS  OP  THE  MOON.  [67,68 

We  thus  liuve, 

luni-.s«)lar  precossitm  =  1*  +  P' 

f,     the  obli(|iiity  ()t"  the  ecliptic; 

/<,    the  ratio  of  the  masH  of  the  Moon  to  tliat  of  the  Earth; 

A,  the  nu'iin  luomciit  of  inertia  of  tlie  Earth  rehitivo  to  axes 

pa.s.siny  througli  its  eijuator; 
C,   the  same  moment  relative  to  its  polar  axis. 

With  these  iletiiiitions  we  have, 

(ieiK^ral  viiliio.  ."^in'cial  viiliio  for  1850. 


/'     0 


u     ( '  -  A 


N  =  [r).4(>289|    cos  6  ,   ''     y~~  =  io.aG.")4l'|    ,   '-  "  ~ 

/«     C  -  A  ^  J o.»;{75S5 1  J.L.  ^'  -   "" 
'  '  1  +  /<      C 

('- A 


r  =[."».!)7r)052]cos  f 


l  +  ;<      C 


P'  =  [;5.725()!)]    cos  f  ^  ~  ^^  =  |3.(J87(52] 


C 


The  special  values  for  1850  are  fouiul  by  putting  for  the 
value  of  the  oblicpiity  of  the  ecliptic  for  18.^0, 

£  =  22°  27'  31".7 

The  )uasfi  of  the  ^foon  from  the  observed  constant  of  nutation. 

08.  From  the  two  quantities  given  by  observation    \  and 
P  +  P'  =ih\,  these  equations  enable  us  to  determine  the  two 

(J A 

unknown  quantities  /.4  and    -  /i     ••   -^^^    the    easiest  way  of 

showing  the  uncertainty  of  the  Moon's  mass,  arising  from 
uncertainty  of  the  precession  and  nutation,  I  give  the  value  of 
its  reciprocal  corresponding  to  ditt'erent  values  of  these  quan- 
tities in  the  following  table: 

Reciprocals  of  the  mans  of  the  Moon  corresponding  to  different 
values  of  the  nutation-constant  and  J  mii-solar  precession. 


/o 

N:=9'^20 

N:=9'^2i 

N 

1 
-9''.  22 

// 

.   50. 35 
50.36 
50.  37 

81.81 
81.86 
81.91 

81.53 
81.58 
81.63 

81.  25 
81.30 
81-35 

C8,  l»yj  THE  CONSTANT  OF  ABERRATION.  133 

Taking  for  the  constaut  of  nutation  the  value  Just  found, 

y  =  9".L'10  L  "MS 

and  for  the  luni-Hohu'  ))re(!e«sion, 

p^  =  .■iO".3<i  I   ".(MM) 

we  iiave,  for  the  reciprocal  of  the  uniss  of  the  ^loon  aud  its 
mean  error: 


11 

bf 
11- 

M 


^  =  81.58  i  (K20 
M 

The  CoHxtaut  of  Aberration. 

09.  In  the  determination  of  astronomical  constants  the  inves- 
tigation of  the  constant  of  aberration  necessarily  takes  a  very 
important  place,  not  only  on  its  own  account  but  on  account  of 
its  intimate  connection  with  the  solar  parallax.  X  general 
determination,  founded  on  all  the  data  available,  was  therefore 
commenced  by  me  as  far  back  as  1800,  before  the  fact  of  the 
variation  of  terrestrial  latitudes  had  been  well  established. 
The  successive  discoveries  of  the  law  of  this  variation  by 
Chandleu  required  such  alterations  in  the  work  as  it  went 
along  that  nuich  of  it  is  now  of  too  little  value  for  publicjition 
in  full.  Happily  the  necessity  for  a  new  discussion  of  the  best 
determinations  at  Pulkowa  has  been  done  away  with  by  the 
papers  of  Chandleu  himself  in  the  Astronomical  Journal. 

Quite  apart  from  the  disturbing  intluence  of  the  revolution 
of  the  terrestrial  pole  upon  the  determination  of  the  constaut 
of  aberration,  this  constant  is  itself  the  one  of  which  the  deter- 
mination is  most  likely  to  be  aft'ected  by  systenuitic  errors. 
In  this  resi)ect  it  is  at  the  opposite  extreme  from  the  constant 
of  nutation.  From  the  very  nature  of  the  case  it  re(|uires  a 
comparison  of  observations  at  opposite  seasons  of  the  year, 
when  climatic  conditions  are  ditlerent.  T.i  most  cases  the 
determination  must  even  be  made  at  different  times  of  day. 
The  effect  of  aberration  on  a  star,  for  example,  is  generally  at 
one  extreme  when  the  star  culminates  in  the  morning,  and  at 
the  other  extreme  when  it  culminates  in  the  evening.  The 
culminations  at  opposite  seasons  of  the  year  are  necessarily 


1  ^ 


I 

[I 


184 


THE  CONSTANT  OF  AHER RATION. 


[69 


asHociatcd  with  riiliiiiiiatioiis  at  <»pi)osito  tiiium  of  tin'  «lay. 
Moreover,  ill  observations  to  (leteriiiine  the  constant  of  aber- 
ration from  Declination,  the  stars  wliieh  ^ive  tlio  hiiffest  eoi'tli- 
cients  are,  for  tiie  nortliern  lieinisphere,  tlioso  near  IH''  of  Ki^ht 
Asrensioii.  Any  error  peculiar  to  the  times  or  seasons  at 
which  these  stars  are  observed  will  therefore  affect  the  result 
systematieally. 

Kij>ht  Ascensions  of  close  polar  stars  also  lead  to  a  value  of 
this  constant.  But  the  same  ditlienlty  still  exists.  In  this 
case  the  tnaxima  and  minima  of  aberration  occur  when  the 
star  culminates  at  noon  and  mi<lnight.  Not  only  is  the  aspect 
of  the  star  dilferent  at  the  two  culminations,  but  the  efleet  of 
any  diurnal  clian^ic  in  the  instrument  will  be  transferred  to  the 
final  result  for  the  aberration. 

The  prismatic  method  of  Loewy  is  free  from  some  of  these 
objections.  Hut  its  application  is  extremely  laborious,  and  we 
have,  up  to  the  present  time,  only  two  determinations  by  it, 
one  by  LoEWV  hiin8«'lf,  which  is  only  regarded  as  preliminary, 
and  one  by  Comstock,  in  which  a  largo  uncertain  correcticm 
for  i)ersonal  e(| nation  was  applied. 

Under  these  circumstances  the  seeking  of  results  derived  by 
nietliodsof  the  greatest  possible  diversity  is  yet  more  strongl 
recommended  than  in  the  <Mse  of  tin?  other  astronomical  co;. 
stants.  I  have  therefore  us<mI  not  only  the  Pulkoava  deter- 
minations, but  all  those  made  elsewhere  which  it  seemed  worth 
while  to  consider.  Notwithstanding  the  great  amount  of  mate- 
rial added  to  Nyrkn's  i)aper  of  LSSS,  it  will  be  seen  that  the 
l)robable  error  of  the  final  result  at  which  I  have  arrived  is 
greater  than  that  which  he  assigns  to  his  result.  This  is  a 
natural  conseipience  of  combining  so  many  separate  determi- 
nations. The  advantage  is,  however,  that  the  assigned  prob- 
able error  is  more  likely  to  be  the  real  one.  It  is  not  to  be 
supposed  that  any  of  the  systematic  errors  already  indicated 
would  pertain  to  all  observers  and  to  all  instruments.  The 
final  outcome  should  be  a  result  in  which  the  discordances  of 
the  separate  determinations  show  the  probable  values  of  all 
the  actual  errors,  both  accidental  and  systematic. 

Determinati<ms  founded  on  the  Eight  Ascensions  of  circum- 
polar  stars  are  not  afiected  by  the  motion  of  the  terrestrial 


Li 


60,  70] 


THE  CONSTANT   oK   ABEHHATION. 


13.> 


iixis,  nor  aro  thoso  foiiiulod  on  (h'rlinations  of  tlioso  Hfais,  if 
only  tli«^  (U'clinatioiis  an*  observ*'*!  ('(|Hiilly  at  Ixttli  «'iiliniiia- 
tion.H.  I>ur  (Ictei'iniiiatioiis  fonn<le<l  on  declinations  of  stars 
fioin  upin'i"  culmination  only  aic  lUMcssarily  alViM'ted  by  this 
cause.  If  however  the  stais  on  which  the  <l«'terinination  is 
based  extend  thronjfh  the  whole  circle  of  Hiyht  Ascension  the 
ott'eet  of  tin'  cause  in  question  may  be  wholly  eliminated  by  a 
suitable  treatment  of  the  e(|  nations  of  coml  it  ion.  To  practically 
eliminate  the  injurious  I'lVect  it  is  iH)t  even  necessary  to  deter- 
mine th(!  exact  law  of  variation.  In  t\wi,  if  the  stars  observed 
ar«^  e(iually  scattered  in  Uiyht  Ascension,  the  effect  of  the  varia 
tion  will  be  partially  eliminated  without  taking-  account  of  it. 

CilANDl-KR  has  shown  that  there  are  two  periodi<'  terms  in 
the  variation  of  latitude,  om^  having;  a  period  of  one  year,  the 
other  of  foui'  hundred  and  twenty  seven  days.  I  n»ay  remark 
that  this  combination  is  in  accord  with  my  theory  developed  in 
the  Motithhj  Xoiiccs  o/  the  Ifoi/ol  AnfroHomical  Sorirfi/  for  March, 
ISOL'.  It  was  there  siiown  that  any  minuti^  annual  (dmnjic  of 
the  position  of  the  principal  axis  of  inertia  (tf  the  lOartli — a 
chanjic  which  mi};ht  be  pnxluc  1  by  tlui  uu>tion  «)f  water,  ice, 
and  air  on  its  surface — W(»uld  appear  as  an  anumd  tei-m  in  the 
latitude,  six  times  as  yreat  as  its  actual  amount. 


n 


1)6 


lie 


111 


Valtns  of  the  constant  of  aberration  (terireil  from  ohner  rations. 

10.  What  I  hav<'  done  since  this  discovery  by  Ciiam)Li:r 
has  been  to  rcexanune  the  detei-minations  of  the  constant  of 
aberration  made  from  time  to  time,  to  make  sncli  conections 
in  their  bases  as  seemed  necessary,  and  more  especially  to 
determine  the  correction  to  be  applied  to  each  sepaiate  result 
on  account  of  the  periodic  term  in  the  latitude.  No  attempt 
was  nnule  to  rework  completely  the  original  material,  except 
in  the  case  of  the  results  of  the  I'ulkowa  and  Washington 
observations  with  the  prime  vertical  transit.  In  the  case  of 
the  former,  however,  the  preliminary  results  reached  from  time 
to  time  were  so  accordant  with  those  of  Chandler  that  it  is 
a  nmtter  of  indifference  whether  we  regard  them  as  belonging 
to  his  work  or  to  my  own. 

Owing-  to  the  very  ditierent  estimates  placed  by  the  astro- 
nomical world  upon  the  Pulkowa  determinations  and  those 


L— WUM 


136 


THE   CONSTANT   OF  ABERRATION. 


[70 


I 


m 


ml 

Is  i :  1'  f 


b 


It-.! 


made  elsewhere,  I  have  used  the  former  quite  apart  from  the 
others.  The  complete  discussion  of  each  separate  value  is 
too  volununous  for  tlie  present  publication,  and  is  therefore 
reserved  for  a  more  exteuded  future  publication.  At  pres- 
ent it  appears  sufHcient  to  judge  the  final  result  by  the  general 
discordance  of  the  material  on  whicli  it  rests,  rather  thau  by 
a  separate  criticism  of  each  i>ai'ticular  cmse. 

In  the  exhibit  of  results  which  follows  it  is  to  be  remarked 
that  NYUE^'s  prime  vertical  observations  do  not  receive  a 
weigiit  as  great,  relative  to  the  other  Pulkowa  determinations, 
as  would  be  given  by  their  assigned  probable  errors.  The 
reason  of  this  course  is  that  one  can  not  be  entirely  confident 
that  the  results  of  any  one  observer  wi*h  this  instrument  are 
free  from  constant  error  arising  from  differences  of  personal 
equation  in  observing  a  bright  and  a  faint  star.  Many  of  the 
Pulkowa  observations  are  'lecessarily  made  in  the  morning  or 
evening  twilight.  In  the  case  of  an  evening  observation  the 
star  will  therefore  be  much  fainter  on  account  of  daylight 
when  it  transits  over  the  east  vertical  thau  it  will  when  it 
transits  over  the  west  vertical  one  or  two  hours  later.  In  the 
case  of  morning  observations  the  reverse  will  be  true.  It  is 
easy  to  see  tluit  if,  in  consequence  of  this  diU'erence  of  aspect, 
the  observer  notes  the  passage  of  the  faint  image  too  late,  the 
-effect  will  be  to  make  the  constant  oi  aberration  too  large. 
The  existence  of  this  IV. 'in  of  personal  e<iimtion,  when  transits 
are  recorded  on  the  chronograph,  is  so  well  known  that,  had 
N YUEN'S  observations  been  made  in  this  ^T;,iy,  I  should  not 
have  hesitated  to  ascribe  the  large  values  of  his  aberration 
constant  to  this  cause.  Although  it  has  never  been  shown 
tliat  any  such  personal  ecjuation  exists  when  observations  are 
made  by  eye  and  ear,  as  Xyken's  were,  yet  when  we  consider 
that  we  are  dealing  with  quantities  amounting  only  to  one  or 
two  hundredths  of  a  second  of  arc.  and  tliat  a  personal  etpia- 
tiou  of  this  kind,  undiscoverable  by  ordinary  investigation, 
might  aftect  the  result  by  this  minute  amount,  we  can  not  but 
lia\  e  at  least  a  suspicion  that  his  values  may  be  slightly  too 
large  from  this  cause. 


T 


t 


701  THE   CONSTANT  OF  ABERRATION.  137 

Separate  results  for  the  constaut  of  aberration. 

A.  iStaiitlard  Pulkowa  determiuat'ous : 

Ah.       tri. 

Observations    with    Vcitical    Circle ;    Polaris,    by       ,^ 
Peters .    20.r>i     2 

Observations  with  Verticil  Circle;  7  niis<"~lia  leoiis 

stars,  by  I*eters 20.47      2 

Observations  with  Vortical  Circle;  18G;J-1870,  Po- 
laris, by  Gylden 20.  tl     2 

Observations  with  Vertical  Circle;  1S71-1S75,  Po- 
laris, by  Nyren 20.51     2 

Observations  with  l*riuie   Vertical;   1S42-1844,  by 

Struve 20.48      4 

Observations   with   Prime  \'ertical;   1.S70-1880  by 
^^YRl:N 20.52      0 

Obsc'.vations  with  Prime  Vertical;    187.VIS70,  by 
Nyren 2n.r).{     1 

Observations  with   ^'ertical  (^Urcle;    180;>-IS7.">,  by 
Gylden  and  Nyren 20.52     2 

Wagner:  Transits  of  three  polar  stars      ....     20.48     5 

From  Pight  Ascensions  of  Polaris;  lS42-lcS44,  by 
LiNDiiAaEN  and  Sciiweizer 20..">0     2 

Mean  reault:  20".49;{  L  0".011 

Tliis  residt  nniy  be  regardeii.  as  identical  with  that  fonnd  by 
Nyren  in  1882. 

B.  Other  determinations: 

Ah.  e  ut. 

AmvERS,    from    observations    witli    the        ,, 

zenith  sector  at  Kew l;(»,.5.'i      [.12        0.5 

ArwERS,  from  Wansted  observations   .     20.4<»     .4z.l2        (»,5 

Peters,  from  Pradley's  obscrv;itions 
of  y  Draconis  at  Greenwich  with  zenith 
sector,  1750-1754 20.07  0.5 

Bessel,  from  Pijiht  Ascciisioh  i  observed 
by  Bradley  at  Greenwich     ....    20.71     L.071      0.5 

LiNDENAU,  from  Pight  Ascensions  of 
J*olaris  observed  at  various  observa- 
tories between  1750  afid  1810  .     .     ,    .    20.45     ±.05        3 


n 


138  TJIE   CONSTANT   OF   ABERRATION. 

tSrjtarate  n-nults  for  the  nnisUtnt  of  dhet't'tttioH 

J>.  Otlier  (k't(  riiiiiiJitioii.s — Continued. 

I)RIM\LE\ ,  from  ^.^^sel■viltion.s  of  thirteen  "'• 

stars  at  Trinity  C<>llej;« ,  J)ublin,  with        ^, 
the  S  foot  ciide 20.40 

Peters,  I'roni  Sira  vi:\s  J)orpat  observa- 
tions of  six  jjairsofcircunipohir  stars   .     L'(>..'3(* 

l{I(•lIARI)so^,  i'roni  observations  witli  tlie 
(1  reel! wieh  innral  circles L'O.oO 

PetkK'S,  from  Kiglit  Ascensions  of  Polaris 
at  Dori.at 20.41 

IjUNDAHL,  fi'om  Declinations  of  P<>l:iris 
at  Dorpat 20.05 

HiiNDHRSoN  an«l  M<"Lear.  from  /»'  and 
(\'^  Centauri I'O.o'J 

Main,  from  observati(tiis  with  the;  (irreen- 

wici:  zenith  tnbe 20.20 

J>()WNlN(l,  from  observations  of  /\J)ra 
eonis  with  reth'x  /.eiiith  tube    ....     20..~)2 

Xkwcomp.,  IVom  observations  <»f  ^fLyra* 
with  th(^  \\  asliiuffton  lU'ime  vertical 
transit,  18<J2-18(;7   ........     20.4<J 

NE\V('(»:Mn,  from  Right  .\sc(Hisi(nis  of 
Polaris  observed  with  the  Washinj;ton 
transit  «'ircle,  1800-1807  .......     20.5.5 

Ki'STNi'-R,  from  observations  of  ])airs  of 
stars  by  the  Ta'jott  method      .     .     .     20.40 

Pkkston,  from  observations  with  the 
Talcott  method  at  Honolulu,  1801- 
1802 20.43 

LoEWV,  fr(»m  his  ]u'ismatic  method     .     .    20.4."> 

CoMSToCK,  using-  LoKwvV  method, 
sli-ihtly  modified 20.44 

Ki  STNHR,  from  M  Auoi  se's  observations*, 

188!)-1800 20.40 

Waxacii,  from  Pulkowa  prime  vertical 
observations 20.40 


1 70 


— Continued. 


wt. 


i.io 

I 

-L  .07 

>> 

±.00 

3 

6 

5 

1.10 

1 

i.io 

I 

i  .05 

4 

tO.4  0 

ri  .05  3 

4 


i.05 

4 

J,  .04 

5 

3 

-t.018 

4 

d=  .015 

4 

70,  71j  THE  LUNAR  1N1:C)UAL1TY.  139 

Separate  rtsulU  for  the  cnnstont  of  aberration — Contimied. 
I'..  OtluT  deteriniiiations — (.'(mtimu*(l. 

.lb.         wt. 

From  Greenwich  Kight  Ascensions  of  ])oljir  stars  ,^ 

with  the  transit  circh^ i'0.;{!)       ;5 

BEf;ivEK,  from  ohservations  at  Strasbnr};'  by  the 

Talc'ott  method,  1S1M)-1,S!K3 LMU7       (5 

Davidson,    from    simihir    observations    at  Sun 

Francisco,  1802-1894        20.  J8       (» 

Mean  result  of  IJ:  Ab.  const.  =  20".46;{  -1  0".013 

The  two  results.  A  and  B,  dirter  by  0".0;{0,  a  quantity  so 
much  f»reater  than  their  mean  errors  as  to  leave  room  for  a 
suspicion  <»f  constant  error  in  one;  or  both  means. 

Thr  Lunar  hutinaliti/  in  the  Earth\  motion. 

71.  The  source  of  t- s  ine(|uality  is  the  revolution  of  the 
center  of  the  Earth  ai.'ind  the  <'enter  of  mass  of  the  Karth 
and  ^looii.  The  former  center  describes  an  orbit  which  is 
similar  to  tliat  of  the  Moon  around  the  Farth.  Since  tliis 
orbit  is  not  a  Keplerian  eclipse,  but  is  affected  by  all  tlu'  per- 
turbations of  thi;  ]\Ioon  by  the  Sun.  no  such  element  as  a  semi- 
major  axis  can  be  assigned  to  it.  Instead  of  this  I  take  as  the 
jtrincipal  element  of  the  orltit  tin-  coettlcient  of  the  sine  of  the 
Moon's  mean  e!  )ngation  fron  -nn  in  thr  expression  foi-  the 

Sun's  true  longitude.  This  elcinenr  is  a  tnnction  (»f  th«'  -^ular 
parallax  and  of  the  mass  of  the  Moon  w  hi»  li  may  Ik  ('('ii\cd 
from  the  foUowing  expression.     Let  us  put 

/<;  the  ratio  of  the  mass  of  the  ,Moon   t>>  ihar  oi   the 
Earth ; 
>•,  A, /i^;  the  radius  vector,  true  longitude  and  l;itiiudc  of 
the  Moon ; 
r',\',fi'',  the  same  coordinates  of  the  Sun; 

.s;  the  linear  distance  of  tiie   Earth's  center  frouj  the 
center  of  mass  of  the  Earth  and  Moon. 


■M 


I 


I 


il 


140 


THE  LVNAB  INEQUALITY. 


I'l 


We  then  have,  for  the  perturbations  of  the  Suu'8  geocentric 
place  due  to  the  cause  in  question : 

J  loy  r'  =  *,  cos  (i  cos  {\--X') 
J\'  =  *  cos  /^ain  (\-A.') 


J/3'  =  '  sin  /i 

r 


and 


/<     r 

1  -f  7<  »^' 


I  have  developed  those  exiuessions,  putting 

TTo  =  .S".H48 


/<  = 


81 


and  taking  for  the  Moon's  coordinates  the  values  found  by 
Delai'NAY.     Putting 

D;  the  mean  value  of  A— A.' 
(f,  (f ;  the  mean  anomalies  of  the  Moon  and  Sun,  respectively, 
v';  the  Sun's  mean  elongation  from  the  Moon's  ascending 
node ; 


the  result  for  JA'  is 

JA'  = 


(».533  sin  D 
+  0.013  sin  3  D 
+  0.179  sin  (D  +  fj) 
-0.4L»l)sin  (I)  -f/) 
+  0.174  sin  (D  —g') 
-0.0(51  ,vin  (I)  +  r/') 
+  0.030  sill  (3D  — (/) 
-O.OU  sin  (D  -(J -.(/') 
—  0.013  sin  2  «' 

This  value  of  the  lunar  inequality  is  substantially  uientical 
with  that  computed  from  the  tables  and  formuhe  of  Lever- 


711 


THE   LUNAR    INEliUALlTY. 


141 


BIEB's  solar  tables.  The  development  of  the  niunbers  there 
given  lead  to  the  value  (»".5;J4  of  the  principal  eoetticient. 

We  have  now  to  find  what  valne  of  the  coetlicient  is  given 
by  observations.  The  observations  I  make  use  of  are  (1)  all 
the  observations  of  the  Sun's  Right  Ascension  from  early  in 
the  century  till  1804;  (U)  The  heliometer  observations  of  Vic- 
toria made  in  1881)  on  Gill's  i)lan  and  worked  up  by  him. 

I  had  intende<l  to  use  all  tlie  observations  of  the  Sun  up 
to  the  present  time.  I  found  however  that  those  made  after 
1804  gave,  by  comparison  with  the  published  ephemerides, 
inadmissible  positive  corrections  to  the  coefficient.  This  cii'- 
cumstance  gives  rise  to  a  strong  suspicion  that  in  the  process 
of  interpolating  the  Right  Ascenshms  ot  the  Sun  during  at 
least  some  years  after  1804,  the  inequality  in  (juestion  was 
rounded  oft"  to  the  amount  of  several  hundredths  of  a  s«'cond. 
The  results  were  therefore  entirely  omitted. 

The  results  for  previous  years,  when  the  inequality  was 
computed  separately  for  every  day  of  obser\ation,  are: 


//P 


wt. 


Greenwich, 

18l'0-'04; 

-.008 

3.0 

Paris, 

1801-'«J4; 

-  .050 

0.8 

Konigsburg, 

1820-'45; 

-.054 

1.2 

Cambridge, 

18L'8-'r>8; 

-  .047 

2.0 

Dorpat, 

1823-'38; 

4-  .100 

0.3 

Pulkowa, 

1842-'04; 

-  .058 

0.5 

Washington 

1840-04; 

.000 

0.2 

Mean, 

JP  =  -  0".048  i-  0".01S 

Gill's  result  is  given  in  the  Monthly  Xofices,  lioi/al  Astro- 
uomieal  Society,  for  April,  1894  (Vol.  LIV,  page  3.")0.)  It  is 
derived  in  the  following  way.  In  the  solar  ephemeris  which 
he  usyd  for  comparison  the  lunar  inequalities  were  computed 
rigorously  from  the  coordinates  of  the  Moon,  putting 

7r  =  8".880 
;/  =  1  ^  83 

To  the  coefficient  P  thus  arising  he  found  a  correction, 

JP  =  +O".O40 


142 


THE   LUNAR    INE(,>rALITY. 


|71,72 


h 


I 


The  above  valiU'S  (»f  rr  aiul  /<  yiv*',  on  tlu^  tlifory  .just  devel- 
oped, 

V  =  (I".  K»0 
Thus  Gill's  result  is,  in  etlect, 

V  =  (;".44(; 

whih'  Miine,  from  obfiervations  of  the  Sun,  is 
(;".53;?  —  ()".04S  =<>".48ri 

I  consider  that  these  results  are  entitled  to  eijual  weij^ht,  an«l 
that  we  may  take,  as  the  result  of  observation, 

p  =  (;".4<;r»  rt  (►".oir. 

Soior  paralht.r/roin  the  lunar  iiiciiuality. 

72.  With  the  mass  of  the  Moon  already   found   from   the 
observed  coustant  of  nutation, 

;/  =  1  :S1.5S  (1  i  .OOlio) 

we  niiiy  now  derive  a  value  of  the  solar  parallax  (piite  inde 
pendent  of  all  other  values.  The.  relation  between  P,  tt,  and 
the  mass  of  the  Moon  is  of  the  jjfeneral  form 

where  k  ia  a  numerical  constant,  and,  for  brevity, 

We  have  found  that  the  following  values  <'(M'res]»ond  to  one 
theory : 

TT  =  8".848 ;         //'  =  82 ;         V  =  iV'.b'Sd 

Hence  follows 

log  /.•  =  1.78207 

BO  that  we  have 

/y'P=  [1.78207]  TT 

The  numerical  values  P  =  6".4(m  and  /«'  =  82.58  now  give 

;r  =  8".818±0".030 


li 


72 


73J 


PAKALLAX  FROM  TRANSITS   OF  VENIS. 


143 


1(1 


IV 


e 
il 


Values  of  the  solar  paraUa.v  derirol  from  meamrenun  ts  of  Venus 
on  the  fare  of  the  iSun  dnrUuj  the  traimits  of  1ST4  and  188:^, 
with  the  heliometer  and  photohelioijraph . 

73.  I  put  these  (leterinination.s  into  one  class  because  they 
rest  essentiiilly  on  the  same  ]M-iiicipIe.  lioth  consist,  in  eftect, 
in  measures  of  tlie  distance  between  the  center  of  Venus  and 
tlie  center  of  the  Sun;  the  latter  being  defined  through  the 
visible  limb.  The  method  is  therefore  subject  to  this  serious 
drawback :  that  the  parallax  depends  ui)on  the  measured  (lifter- 
euce  between  arcs  which  nniy  be  from  thirty  to  fifty  times  as 
great  as  the  parallax  itself,  the  measures  being  made  in 
different  parts  of  the  earth. 

The  equations  of  <'ondition  given  by  the  American  photo 
graphs  of  1374  are  found  in  Part  I  of  Observations  of  the 
Transit  of  Venus,  December  U,  1874;  Washington,  (jovernmeut 
Printing  Oflice,  1880.  A  preliminary  solution  of  these  ecpia- 
tions,  the  only  one,  however,  to  which  they  have  yet  been  sub- 
jected, was  published  by  D.  P.  Todd,  in  the  Ameriean  Journal 
of  Seience  for  .lune,  1881.    (Vol.  XXT,  page  4!)0.) 

The  photographs  of  1882  have  been  completely  worked  up  by 
Professor  IIaukness,  and  the  results  are  found  in  the  Report 
of  the  Superintendent  of  the  Naval  Observatory  for  1880.  The 
ecjuations  derived  from  the  German  heliometer  measures,  with 
a  preliminary  discussion  of  their  results,  are  officially  published 
by  Dr.  AuwEUS,  in  the  Bericht  ilber  die  deutsehen  lieobachtuuyen, 
V,  p.  710. 

The  sepiirate  results  for  the  parallax,  with  the  probai)le 
errors  assigned  by  the  investigators,  are  as  follows: 


1874: 


n 


1882: 


Photographic  distances, 
Position  angles, 
Measures  with  heliometer, 
Photographic  dist?,nces. 
Position  angles, 
Measures  with  heliometer, 

Under  w  is  given  a  system  of  weights  proportionally  deter 
mined  from  the  probable  errors  as  assigned.    Using  this  sys- 
tem, the  mean  result  is — 

n  =8".854  ±  ".010 


,1 

IC. 

w 

8.888  ±  0.040 

6 

1 

8.873  ±  0.060 

3 

3 

8.87()  ±  0.042 

5 

5 

8.847  i  0.012 

04 

6 

8.772  ±0.050 

4 

4 

8.871)  i  0.025 

10 

10 

lU 


PARALLAX  FROM   TRilNSITS  OP   VENUS. 


173 


it 

If 


I  ('(uiceive,  liowever,  that  these  rehitive  weights  <lo  not  eor- 
respoml  to  the  actual  precision  of  the  measures.  Tlio  very 
small  probable  error  assigiKMl  by  Prof.  FIaukness  to  the  result 
of  the  photographic  distances  of  1882  does  not  include  the 
juobable  error  of  the  angular  value  of  the  unit  of  distance  ou 
the  jdate,  which  may  arise  from  a  number  of  sources,  includ- 
ing the  possible  deviatiou  of  the  mirror  of  the  instrument 
from  a  perfe«t  ]>lane.  From  this  error  the  positi«m  angles 
are  entirely  free.  I  have,  therefore,  assigned  another  set  of 
weights,  w',  which  seem  to  me  to  correspond  more  nearly  to 
the  facts.    The  result  of  this  system  is — 

rr  =  8".857  -1-  ".016 

This  mean  error  is  derived  from  the  individual  discordances, 
and  n«)t  from  i'omparisons  with  the  vahies  of  tlie  parallax 
otherwise  determined.  As  there  may  be  a  fortuitous  agree- 
ment among  the  separate  values,  another  estimate  may  bo 
made  on  the  basis  of  the  total  mean  error  derived  by  Auwers, 
which  includes  all  known  sources  of  error.  lie  finds  f  =  i.  ".(►32 
for  the  combined  heliometer  results,  to  which  I  have  Pissigued 
weight  15.    Hence,  for  the  total  weight  20,  we  have— 

e=i  0".02;i 

The  deviation  of  the  above  result  from  the  mean  of  all  the 
other  good  ones  is  worthy  of  special  attention.  The  deviation 
is  more  tlian  three  times  its  mean  error,- and  therefore  between 
four  and  five  times  its  probable  error.  We  must  therefore 
accept  one  of  two  conclusions,  either  the  probable  errors  liave 
been  considerably  underestimated,  or  the  method  is  affected 
with  some  undiscoverable  sourca  of  systemati*;  error,  which 
nuxkes  it  tend  to  give  too  large  a  result.  The  close  accordance 
of  the  six  separate  results,  of  which  only  a  single  one  deviates 
from  the  adopted  mean  by  more  than  its  probable  error,  and 
that  by  only  a  little  more,  would  give  color  to  the  view  that 
the  err«»r  is  a  systematic  one,  and  that  through  some  unknown 
caui^e  Venus  is  always  measured  too  low  relatively  from  the 
center  of  the  Sun.    1  can  not,  however,  think  of  any  such  cause. 

If  we  determine  the  mean  error  from  the  deviations  of  the 
separate  results  from  what  we  know,  in  other  ways,  to  be 


74| 


PARALLAX  FROM  TRANSITS  OF  VENUS. 


145 


Koaiiy  the  most  probable  value  of  the  parallax,  namely  8".80, 

we  have — 

■  // 

Mean  on  ror  to  weight  1 ;    I  .1 48 

Mean  error  of  result  I:  .0-*.> 

Solar  itarnUux  t'fnm  nhserreil  coHtnctn  ilurinij  transitu  of  VenU8, 

74.  The  contact  observations  of  17«»1  and  17<H>  are  discussed 
in  Astr<nu>nii<al  Papfrs,  Vo\.  III.  I  have  also  ninde  a  coni- 
]dete  discussion  of  those  of  1874  and  1882,  \vhi(di,  at  the  date 
of  writinj;-.  is  unpublished.  The  separate  results  frouj  each 
contact  follow. 

In  the  case  of  the  second  contacts  of  1874  and  188L'  it  was 
found  ne<'essary  to  divide  the  observations  into  two  classes: 
those  of  mean  or  true  conta«!t,  and  those  of  tiie  formation  of 
the  thread  of  light.  In  the  case  of  thi;  third  contact  no  such 
division  was  necessary,  as  the  observations  c<uild  generally 
be  referred  to  the  same  mean  phase.  The  mean  error  which 
follows  each  result  is  derive«l  from  the  discordance  of  the 
separate  observations. 

Valuen  of  the  nolar  parallax  from  ohserred  contacts  of  the  limb 
of  Venus  with  that  tfthe  Hun. 


1761        III;   7r 

=  8.78  i 

.12;     1 

r.  =     8 

IV; 

8.75  ± 

.20 

3 

1709,          I; 

0.04  ± 

.17 

4 

II; 

8.55  i 

.13 

7 

III; 

8.72  rk 

.09 

14 

IV; 

9.01  ± 

.12 

8 

1874,          I; 

8.95  i- 

.24 

2 

11;  M; 

8.78  i 

.061 

30 

11;  L; 

8.75  i 

.10 

11 

III; 

8.76  i 

.045 

57 

IV; 

8.74  ± 

.09 

14 

1882,          I; 

8.93  i 

.15 

5 

II;  M; 

8.76  ± 

.042 

64 

II;  L; 

8.72  rjr 

.072 

.  22 

III; 

8.88  i 

.042 

64 

IV; 

9.07  zL 

.12 

8 

5690  N  ALM 10 

146 


PARALLAX  FKOM  TIIANSITS  OF  VENUS. 


[74 


i:^ 


II 


Tlie  weights  assigned  are  determined  by  these  laeaii  errors, 
taken  on  such  a  scale  that  unity  is  the  weight  for  mean  error 
i  ".330.    Tiie  mean  result  of  the  whole  series  is 

IT  =  8".707  -i:  ".0L>3 

This  mean  error  is  that  resulting  from  the  deviations  of  the 
sixteen  separate  results  from  the  general  mean,  which  give  for 
the  mean  error  corresponding  to  weight  unity, 

f,  =  ±  ".42. 

The  excess  of  this  mean  error  over  that  determined  from  the 
equations  themselves  shows  that  the  general  discordance  of  the 
several  contacts  is  somewhat  greater  than  would  be  inferred 
from  the  individual  discordances  of  the  contacts  iuter  ae.  This 
is  what  we  should  expect  from  constant  errors  in  the  determi- 
nations of  i)arallax  from  each  separate  contact.  I  conceive, 
however,  that  such  constant  errors  are  not  likely  to  be  large; 
and  we  can  not  conceive  that  contact  observations  in  general 
are  subject  to  any  constant  error  tending  to  make  the  parallax 
derived  from  them  always  roo  great  or  too  small.  I  conclude, 
therefore,  that  the  mean  err^r  determined  from  the  totality  of 
the  results  may  be  regarded  as  real. 

It  will  be  interesting  to  compare  the  separate  results  of 
internal  and  external  contacts.    They  are 

//  // 

From  internal  contacts ;    n  =  8.776  ±  .023 
From  external  contacts ;    tt  =  8.908  ±  .00 

These  meau  errors  are  those  derived  from  the  concluded 
results  and  they  show  that  the  exteriml  contacts  are  relatively 
more  discordant  in  proportion  to  the  weights  assigned  than  are 
the  internal  ones.  If  we  consider  this  discordance  to  indicate 
a  larger  meau  error,  and  therefore  assign  a  proportionally 
smaller  weight  to  the  results  of  external  contact,  we  have,  for 
the  concluded  result, 

7T  =  8".791  ±  ".022 

As  these  two  hypotheses  seem  about  equally  probable,  I  shall 
adopt  the  mean  result, 

7r=:8".794 


75]  PARALLAX  FRO 31  VELOCITY  OF  LIOUT.  147 

Solar  parallax  from   thv  obnerved  vomttant  of  aberration  and 
measured  velocity  of  light. 

75.  The  question  of  tlie  souiidnesH  of  the  proposition  that 
the  aberratioti  i.s  equal  to  the  quotient  of  the  veh)cit>  of  the 
Earth  in  its  orbit  by  the  velocity  of  Ii},'ht  is  too  broad  a  one  to 
he  discu8se<l  here.  I  i-an  only  remark  that  its  simplicity  and 
its  general  accord  with  all  optical  phenomena  are  such  that  it 
seems  to  me  it  should  be  accei)ted,  iu  the  absence  of  evidence 
against  it. 

In  Antronomieal  Paper»,  Vol.  II,  page  202,  I  iiavc  given  the 
following  determinations  of  the  velocity  of  light  in  vacuo  by 
MicHELSON  and  myself,  expressed  in  kilometers  per  second: 

Mkhelson  at  Naval  Academy  in  1870 299910 

MiciiELSON  at  Cleveland,  1882 .  299853 

Nkwcomh  at  Washington,  1882,  using  only  results 

supposed  to  be  nearly  free  from  constant  errors    .  2!>98G0 

Newcomb,  including  all  determinations 299810 

I  have  concluded, 

Velocity  of  light  in  vacuo,  =  2998G0  ±  30  k.  m. 

Taking  as  the  etpiatorial  radius  of  the  Earth  0378.2  k.  m. 
(Clark),  the  following  table  shows  the  values  of  the  constant 
of  aberration  corresponding  to  admissible  values  of  the  solar 
parallax  when  this  determination  of  the  velocity  of  light  is 
accepted. 

Ab.  =  20.40         7T  =  8.8076 

20.47  8.8033 

20.48  8.7990 

20.49  8.794G 

20.50  8.7903 

20.51  8.7859 

20.52  8.7810 

20.53  8.7773 

20.54  8.7730 


148 


•  VHALLACrnc   INE(,>rALITV. 


7"),  70 


Wi^  tliUH  Imv<'  for  tin'  valiU'M  <it'  \\w  .solar  pamilax  r«>siiltiiij{ 
from  tln'  two  values  of  tluj  coiistiint  of  aberration  alruiuly 
derived: 


// 


From  I'nikowa  determinations;  Al».  =  L't>.4!».'{.  rr  =  s.793 

From  miseellan«^ons  determinations;    Al).  =  2t».i(».);  tt  =  S.SOO 


! ,: 


iSolar  parnUnx  fmm  the  partdlaclh'  invijualii}!  of  tJir  Moon. 

7<».  I  Inive  tlorivj'd  a  valneolllie  parallaetie  ine(|naiity  <»♦' tlie 
Moon  from  the  nieridnin  observations  made  at  (Jreeinvicli  and 
Wasldnjjton  sin<'e  1S«Jl*.  The  deternunation  of  this  ineqnaliuy 
is  i)eenliarly  liable  to  systenuitie  error,  owin^  to  the  fact  that 
observations  have  to  be  made  on  one  lind>  of  tho  Moon  when 
the  ineiinality  is  ]»ositive,  and  on  the  other  lind)  whou  it  is 
nefjative.  Hence,  if  we  determine  the  ineqnality  by  the  eom- 
parison  of  its  extrem(3  observed  effects  on  tho  Moon's  longitnde 
or  Rif^lit  Ascension,  any  error  in  the  adopted  semidiamet'U"  of 
the  Moon  will  atfoct  the  result  by  its  fall  amonnt. 

It  <loes  not  seem  jnaeticable  to  nnikt  a  reliable  deternuna 
tion  of  the  Moon's  diameter,  beeau;".-  it  will  necessarily  be 
made  near  the  time  of  full  ]\loon,  when  the  illnnunation  ()f  the 
extreme  lind)  is  less  intens*;  than  near  the  (puidratures,  an<l 
when  some  jMirtions  of  the  limb  that  nught  be  visible  if  it  were 
illnndnate<l  by  a  perpemlieular  Sun  will  be  thrown  into  shadow 
by  the  hori/.«mtal  one.  For  these  reasons  it  may  be  expected 
that  the  parallactic  inequ.ality  deternuned  by  asinj;  observed 
semi<liameters  of  the  Moon  will  be  too  large.  I  have  therefore 
adopted  the  plan  of  deternuning  the  inecpiality  from  each  limb 
separately.  To  show  in  regular  progression  the  errors  depend 
ing  on  the  elongati(ni  from  the  Sun,  I  have  classified  the  resid- 
uals <d' observations  ac<'ordlng  to  the  hour  of  mean  time  at  which 
the  Moon  passed  the  meridian;  and  formed  equations  of  con- 
dition i'ontaining  two  nidvuown  (juantities,  the  one  a  constant 
correction  dei)ending  on  the  seuiidiameter,  jjcrsonal  equation, 
etc.,  and  the  other  tho  parallactic  inequality.  The  (juestion  is 
further  complicated  by  the  fact  that  the  majority  of  observa- 
tions near  are  (piadratures  made  during  daylight,  when  it  is 
to  be  expected  that  the  illumination  of  the  atmosphere  will 


■• 


70]  PAUAI.LAc  Tir   INEt,»UALlTY.  1  |f> 

diiiiinisli  the  irnulia(i<ni,  and  tlius  ]v,u\  to  u  sinalh'i  appan'iit 
HtMuidiaiiu'ttT.  I  Imv*'  tlu'ieloie  Moujflii;  to  (h'tt'itiiiiic  loi  tin- 
two  ohstMvatorirs.  hy  a  <'oiii|>arisoii  of  tlu'  uhsnvatioiis.  the 
4'oiTi'cti«Mi  to  Im'  appliod  in  order  to  ri'iliur  oltsnvatioiis  made 
during  dayli^lit  or  t\vili;;lit  to  what  tliry  would  liave  hvm  had 
tile  sky  not  bei-ii  ilhiininuted.  The  reduction  was  sniaMrr  than 
I  had  expected,  and  somewhat  <louhtfnl;  I  liave  assijincd  pro- 
portionally less  \veij,'lir  to  thosi'  observations  when-  it  was 
necessary.  The  followin;;'  ire  the  eipiations  of  condition  thus 
fornuMl.    The  nnknt>wn  tjuantitics  are — 

X,  a   constant,   depending-   (»n    the   semidianicter.    p«'rsonal 

eipmtion,  etc.; 
y,  the  eorreetion  to  the  parallactic  inefpiality  of  the  .M(ton 

after  reduction  to  the  value  8".S4«  of  the  solar  jtarallax. 

(JliKKNWIcn. 
Li  nth  I, 


t 


'" 


h 

// 

4.(;; 

.r-f  0.JK3.J/ 

=  -0.53; 

irt.     0.2 

5.6 

0.00 

-  0.72 

o.i; 

6.5 

0.00 

-  0.11 

1 

7.5 

0.02 

—  0.50 

1 

8.3 

0.70 

—  0.54 

1 

9.5 

0.01 

-0.13 

1 

10.5 

o.;{s 

-  0.00 

1 

11.5 

0.13 

-  o.ot; 

1 

L 

iiiih  II. 

12.5; 

x'-0.13y 

=  +  0.20; 

wt,    1 

13.5 

-  0.38 

-!-  O.IG 

14.5 

-  0.01 

+  0.28 

1 5.5 

-  0.7J» 

+  0.54 

1<>.5 

-  0.02 

-0.1! 

17.5 

^  0.90 

—  0.02 

18.4 

-0.90 

+  0.44 

0.5 

10.4 

—  0.03 

+  1.21 

0.2 

150 


PARALLACTIC  INEQTTALITY. 

WASHINClTOIi. 
TJmb  T. 


[76 


4.0; 

•r  + 

0.03  y  = 

-  1.02; 

ict.  =  0.2 

5.0 

0.00 

-  1 .2(5 

0.4 

0.5 

(».00 

-  0.85 

7.5 

0.92 

-  0.04 

8." 

0.70 

-  0.71 

0.5 

(KOI 

-  0.71 

10.5 

0.38 

-0.48 

11.5 

0.1.i 

Lim 

--  0.23 
h  JL 

12.5; 

x'- 

0.13  y  = 

+  0.?!; 

irt.  =  1 

i;i.5 

— 

0.38 

0.43 

1 

14.5 

— 

0.01 

0.52 

1 

15.5 

— 

0.79 

0.40 

1 

10.5 

— 

0.02 

0.72 

1 

17.5 

— 

0.00 

.     0.00 

0.5 

18.4 

— 

0.00 

1.32 

0.3 

10.4 

^ 

0.03 

J. 50 

0.1 

With  tlu'se  <*(|natioii!S  wo  liavo  our  choice  to  deteriitinc  the 
',5iiranactic  ine(|uality  by  assifjiii'ijif  a  valu"  to  the  seini<liamctcr, 
or  'o  cliiniiiatc  the  semidiaMcter  from  the  iiortnal  c(|uation.s. 
lu  each  case  the  c(iuatioiis  give  the  tbllowiiig  expressions  for  y; 


li! 


Greenwich  :     Limb    i ;  y  =  —  <>.55  —  1.23.r 
"  '^        II;  -0.28+ 1.23 .»' 

Wasliiiiffto  I :  Iamb    T ;  »/  =  -  0.0!)  -  1 .23  ,»■ 
"II;  -0.88+1.20.1' 

If  we  choose  to  ittilize  Mie  observed  diameters  we  liave  the  fol- 
lowing  results: 

From  0(5  transits  of  the  Moon's  diameter  observed  at  (Ireenwich ; 

.!■  -  .»•'  =  -  {y.iw 


6 


76]  PARALLACTIC  INEQiJALlTY.  151 

From  33  transi<^s  observed  at  Washington: 

.»•-.»•'  =  - r'.lL* 

We  sliould  thus  liave, 

// 
Front  Ureenwicili  observations,      y=— 0.02 
From  Wasliington  observations,   //  =  —  0.23 

If,  on     he  other  liaud,  we  eliminate  .r  from  eaeli  pair  of 
normal  ecjuations,  the  final  results  for  y  will  be 

//  //  //     ..,f 

Gieenwich :      Limb    I ;     (M>t  //  =  -  0.4."»;  y  =  —  0.70  4.  O.IO    0 
"       II;     (M;4//=      0.00;  »/=      0.(M>  |   0.30     1* 


n 


Washington :   Limb    I ;     0.04  //  =  -  0.5l»;  y  =  —  O..SI  J:  O.Ki    G 
"  "II;     iirtli  y  =  -  0.32 ;  y  =  -  O.r.0  [:  0.27     3 

The  weighted  mean  of  these  results  is 

.»/=  -0".64  1  0".12 

The  resulting  value  ot  the  solar  parallax  is 

;r  =  o".S02  I   0".00.S 

A  very  careful  determination  of  the  solar  parallax  was  made 
from  the  same  theory  by  Dr.  IJatteuman,  by  meansof  oceulta- 
tions,  and  the  result  is  discMissed  very  fully  in  the  publica- 
tions of  the  IJerlin  Observatory.  J)r.  Battkuman's  definitive 
result  is 

TT  =  8".704  A,  ".010 

I  have  slightly  revised  this  result,  by  applying  a  c(U-reetiou 
to  the  ('oerticient  for  the  parallax  adopted  by  Dr.  Battkrman, 
with  the  result 

T  =  8". 789  :L  ".010 

Accepting  this  result,  and  combining  it  witli  that  ab*eady 
found  from  meridian  observations,  the  parallax  from  this 
method  will  tinally  come  out 

7T  =  8".709  i  ".007 

This  mean  error  may  be  reganled  as  belonging  to  the  doubtful 
class. 


-^''1 


162  SOLAR  PARALLAX  FROM  MINOR  PLANETS.        [70,77 

While  tbis  work  is  passing  tlnongli  the  press  there  appears 
an  important  ]»aper  by  Franz  of  Konigsborg,*  giving  the  value 
of  the  parallactic  equation  derived  from  observations  on  tlie 
lunar  crater  Mnsliug  A.  The  correctiim  to  Hansen's  eoetli- 
cient  is  found  to  be 

-  2".10  i  0".30 

Tlie  corresponding  result  for  the  solar  parallax  is 

8".7G7  ±  0".021 

We  may  combine  the  three  results  for  the  solar  parallax 
tlius : 

Gret!\wicli  and  Washington  meridian  obser-  ,, 

vation.: ;r  =  8.802;  tr  =  6 

Battkkmann  from  oceultations 8.7.S".>;  2 

Franz  from  crater  jl/o*///*^/ A 8.7«»7;  1 

Mean 8.704  ±".008 

Solar  imrallax  from   ohscrraiionft  on  minor  planetn  icith  the 

heliometer. 

77.  The  fact  that  tlie  determination  of  the  parallaxes  of  the 
small  planets  by  comparison  with  neighboring  stars  is  free 
from  the  grave  uncertainty  attaching  to  similar  observations 
of  Venus  and  Mars,  owing  to  tlie  absence  of  a  sensible  disk, 
was  long  since  pointed  out  by  Dr.  Galle.  In  1870  be  pub- 
lished a  discussion  of  observations  on  Flora,  made  at  nine 
northern  obscrvat(UMes,  and  at  the  Cape,  Cordoba,  and  Mel- 
bourne in  the  Southern  hemisjdiere.t    The  result  was 


n  =  8".873. 


An  examination  of  the  residuals  of  the  several  observatories 
shows  that  in  tlie  case  of  at  least  one  of  the  Southern  observa- 
tories there  is  a  systeniatic  diil'ercnce  of  a  considerable  fraction 


*  Astronoinisolie  Xachrlchteii,  Vol.  136,  8.351. 

trdier  oiiit)  Hcstiiiiiimiig  der  Soinifn-l'nrullaxo  aua  corre8pon<lireii«1eQ 
lieobiiclituiigca  des  I'laneteu  I'loru,  iu  October  uud  Novumber  1873. 
ItreHliiu,  MuruHchke  •&.  lU'ieudt,  1875. 


77] 


SOLAR  PARALLAX  FROM  MINOR   I'LANETS. 


153 


of  a  second.    This  fact  seems  to  prevent  our  assigning  any 
appreciable  weijrlit  to  the  final  result. 

In  1874,  Gill,  at  Mauritius,  made  lielionieter  observations 
of  Juno,  east  and  west  of  the  meridian,  with  the  same  object. 
The  result  was  8".7<m,  or  8".Sl,j  when  a  discordant  observation 
was  rejected.  In  this  connection,  only  an  allusion  is  necessary 
to  Gill's  expedition  to  Ascension  in  1877,  made  for  the  pur- 
l)ose  of  applying  the  method  to  Mars  at  the  oi)position  of  that 
year. 

Shortly  afterwards  GiLL  published  in  the  first  volume  of  The 
Observatory  a  very  exhaustive  disi'ussion  of  the  methods  of 
determining  the  solar  i)arallax,  in  which  he  showed  that  heli- 
ometer  observations  of  the  minor  planets,  made  either  at  a 
single  station  not  too  far  from  the  eiprntor,  or  at  two  stations 
in  ilitlerent  hemisplieres,  afforded  a  method  of  measuring  the 
parallax  more  i)recise  than  any  before  applied. 

Ten  years  olajjsed  before  the  ]>lan  was  put  into  operatir  i. 
Then,  in  1889  and  1890,  a  (!oncerted  system  of  observations  was 
made  on  the  three  minor  planets,  Victoria,  Iris,  and  Sappho,  at 
a  number  of  observatories  in  both  hemispheres.  The  observa- 
tions relating  to  Victoria  were  carried  out  most  thoroughly, 
in  that  a  very  careful  triangulation  of  the  stars  of  compaiison 
itifrr  86  was  m.ade  at  the  observatories  which  took  part  in  the 
measures.  The  tabular  data  for  the  reductions  were  supplied 
by  the  office  of  the  BerUmr  Jahrhnck,  aiul  the  reductions 
and  discu'^sion  were  made  by  Gill  himself  for  Victoria  and 
Sappho,  and  by  Dr.  Elk!N,  on  Gill's  plans,  for  Iris.  The 
three  results,  as  comnuinicated  in  advance  of  their  complete 
otlicial  publication,  are 

//  // 

From  Victoria:  tt  =  8.8(10  p.  e.  A:  0.006 
Iris :  8.8l.'r»  p.  e.  ±  0.(M)S 

Sappho:  8.7fM)  p.  e.  J^  0.012 

I  assign  the  resjiective  weights  4,  2,  and  1,  thu.-*  obtaining, 
as  the  final  result  of  this  method, 

7r  =  8".807  ±  0".00G 

I  have  included  in  a  separate  category  Gill's  determina- 
tion  by  Mars,   at  Ascension,  in   1877,   as   jmblislM'd  by  the 


J  r 


154 


UNCERTAINTY  OY   PARALLAX  PROM  MARS.   [77,  78 


ill 

lit 


Royal  Astronomical  Society  {Memoirs  Royal  Antronomical  So- 
ciety, Vol.  XLVI),  for  the  reason  that,  owing  to  the  disk  of 
Mar^,  and  its  reddish  color,  determinations  made  on  it  are 
liable  to  errors  peculiar  to  that  planet,  or  at  least  dift'erent 
from  those  which  might  come  in  in  the  case  of  the  small 
planets. 

Remarks  on  determinations  of  the  parallax  tchich  are  not  used 

in  the  present  discussion. 

78.  In  the  preceding  discussion  are  given  the  results  of 
every  modern  method  of  determining  the  solar  parallax  with 
which  I  am  acquainted,  except  meridian  and  equatorial  obser- 
vations on  Mars.  1  have  not  used  any  of  the  results  derived 
from  this  source,  owing  to  their  large  probable  error,  and 
the  suspicion  of  systematic  error  to  which  they  are  open. 
One  of  these  causes  of  error  is  to  be  found  in  the  red  color  of 
Mars.  This  cause  will  be  pointed  out  and  discussed  very 
fully  in  a  subsequent  section.  Its  effect  would  be  to  make  the 
observed  parallax  too  large.  Since,  as  a  matter  of  fact,  all 
the  determinations  of  Mars  by  meridian  observations  have 
given  a  larger  parallax  than  the  generality  of  other  methods, 
color  seems  to  be  gi-'en  to  this  suspicion.  Apart  from  this, 
the  setting  of  the  threads  of  a  meridian  circle  upon  the  appar- 
ent disk  of  Mars  involves  a  visual  estimate  not  comparable 
with  that  of  the  bisection  of  the  image  of  a  st.ar  by  the  threads. 
Hence,  there  is  a  chance  of  systematic  personal  error  arising 
from  this  source.  The  observations  generally  exhibit  large 
discordances,  which  may  be  attributed  to  one  or  the  other  of 
these  causes. 

It  may  be  objected  to  the  inclusion  of  Gill's  Ascension 
result  that  it  should  be  rejected  for  the  same  reason,  since  the 
color  of  the  planet  would  affect  heliometer  observations  and 
meridian  observations  equally.  I  have,  however,  considered 
it  free  from  the  objection  in  question,  for  two  reasons.  In  the 
first  place,  the  result  is  not  too  large,  but  is,  on  the  contrary, 
the  smallest  of  all  the  accurate  measures.  The  principle  that 
when  a  result  is  open  to  n  strong  suspicion  of  being  affected 
by  a  cause  which  would  cause  it  to  deviate  in  one  direction,  it 
is  logical  to  conclade  a  posteriori  that  the  cause  has  not  acted 


J8 

0- 

)f 
e 

It 
II 


78J 


FKCERTAINTY  OF  PARALLAX  FROM  MAKS. 


155 


if  the  (loviatiou  is  found  *o  be  in  the  other  direction,  may  not 
be  a  perfectly  sonnd  one,  but  I  have  nevertheless  acted  upon 
it.  In  the  next  place  Gill  himself,  as  a  part  of  his  discus- 
sion, compared  the  observations  wlien  Mars  was  at  ditterent 
altitudes,  in  order  to  detennine  whether  the  action  of  such  a 
cause  was  indicated,  and  found  a  negative  result. 

In  1890  an  unsuccessful  attempt  was  made,  at  the  writer's 
request,  by  Dr.  Vv .  L.  Elkin,  to  measure  the  ettect  in  <inc.stion, 
by  placing  a  refracting  prism  of  very  small  angle  over  one  of 
the  halves  of  a  heliotucter  objective,  and  measuring  the  refrac- 
tion thus  produced.  It  was  stipposed  that  the  dispersing 
action  of  the  prism  would  represent  that  of  the  atii»os|)here, 
greatly  magnified.  The  failure  arose  from  the  result  that  the 
apparent  mean  refraction  of  the  star  produced  by  the  prism 
proved  to  be  a  function  of  the  star's  magnitude,  ranging  from 
748".79  for  a  star  of  magnitude  2.55  to  751".0l  for  a  star  of  mag- 
nitude 0.95.  The  reason  seemed  to  be  that  too  powerful  a  ])rism 
was  used,  so  that  the  spectrum  was'quite  sensible;  then,  in  the 
case  of  fjiint  stars,  the  red  portion  of  the  spectrum  was  invis 
ible,  so  that  the  apparent  mean  refraction  wijs  greater  than  in 
the  case  of  the  brighter  stars.  The  mean  of  the  observed 
displacements  of  Mars  was  748".G1,  so  that  it  was  always  less 
for  Mars  than  for  the  stars.* 

An  investigation  of  the  question  whether  the  same  effect  is 
noticeable  in  meridian  observations  fails  to  sliow  any  relation 
between  the  brightness  of  a  star  and  its  refraction.  But  this 
does  not  disprove  the  relation  between  the  refraction  and  the 
color  of  a  star. 

On  the  whole  it  seems  to  me  that,  at  least  in  the  case  of 
Mars,  we  have  here  a  cause  so  mixed  up  with  personal  error 
in  making  the  observations  that  the  objective  and  subjective 
effects  can  not  be  completely  separated. 

•  Aslrotiomicdl  Journal,  Vol.  10,  pjigo  9i'. 


1 


CHAPTER  Vlll. 


I 
t; 


DISCUSSION    OF    RESULTS    FOR    THE    SOLAR    PARALLAX 
AND  THE  MASSES  OF  THE  FOUR  INNER  PLANETS. 

7!>.  We  have,  iu  what  precedes,  fouud  or  collected  uiiie 
separate  values  of  the  i>arallax  of  the  Sun,  by  methods  of 
wliich  seven  may  be  rejj:arded  as  completely  distinct,  in  the 
sense  that  no  one  source  of  error  is  common  to  any  two.  Of 
these  seven  the  two  most  nearly  associated  are  those  which 
utilize  transits  of  Venus.  These  are  similar  only  in  the  sense 
of  resting  upon  a  <leterminati(»n  of  the  relative  i)arallax  of 
Venus  and  the  Sun  duriiijf  the  time  of  a  transit.  But  the 
only  common  elements  which  enter  into  the  determination  are 
the  ratio  of  the  distances  of  the  Sun  and  Venus,  which  is 
<letermined  with  such  certainty  that  we  can  not  regard  it  as 
suoject  to  error.  Tiie  metliods  of  determining  the  jKirallax  in 
the  tw(»  cases  are  completely  distinct  from  the  beginning, 
there  being,  I  conceive,  no  common  source  of  error  att'ecting 
an  observation  of  contact  of  limbs  and  one  of  a  distance 
measured  from  the  center  of  the  Sun  while  Venus  is  in  transit. 

1  have  classilied  as  if  they  were  independent  the  values  of 
the  parallax  which  follow  from  the  Pulkowa  determinations 
of  the  constant  of  aberration,  and  those  which  follow  from  all 
other  determiniitions.  Of  course  whatever  tloubts  may  aftect 
the  theory  of  the  assumed  relation  between  the  ccmstant  of 
aberration  and  the  velocity  of  light  will  eciually  affect  both 
determinations.  I  do  not,  however,  conceive  that  there  is 
any  source  of  error  which  can  affect  both  the  Pulkowa  deter- 
minations of  the  aberration  and  those  made  elsewhere.  The 
two  iouhl  have  been  combined  so  as  to  give  a  single  result 
of  the  method;  but  as  the  two  values  of  the  constant  differ 
by  more  than  we  should  expect  them  to  from  their  probable 
errors,  I  have  kept  them  separate,  partly  not  to  give  a  false 
appearani'c  of  agreement  of  results,  and  partly  to  facilitate 
the  inception  of  any  future  investigation  on  the  subject. 
166 


79] 


THE   SOLAR   I'ARALLAX. 


157 


1  liave  also  separated  tlie  result  of  ( I  ill's  observations  on 
Mars,  at  Ascension,  in  1877,  from  tin'  determinations  made  by 
the  same  method  on  the  minor  planets,  because,  owin<;  to  the 
color  and  disk  of  Mars,  the  two  results  may  be  alVerted  by 
very  dlHerent  systenmtic  errors.  The  only  common  systematic 
error  which  seems  likely  to  atlect  them  is  that  arising  from  the 
color  of  the  object,  which  will  be  discussed  hereafter. 


Results  of  (lefenninatious  of  the  sohtf  parallax  arramjeil  in  the 

order  of  maynltude. 

From  the  mass  of  the  Earth  resulthtff 

from    the   secular    rariations  of  the      ^^  ^,  tot. 

orbits  of  the  four  inner  planets  .  .  .  8.7")!)  j  .OK)  9 
From  Gill's  observations  of  Mars  at 

Ascension 8.780  |    .020        2 

From  l*nlh)n-a    determinations    of  the 

constant  of  alterration 8.70;{  -i  .0040    40 

From  observations  of  contacts  lUirimj 

transits  of  Venus 8.794  i  .018        3 

From  the  parallacfie  inequaliti/  of  the 

Moon 8.794  ,t:  .007      18 

From  determinations  of  the  constant  of 

aberration    made  elsewhere   than  at 

Pulkon-a 8.800  ±  .005(5    28 

From    heliometer   (d>servations    on   the 

minor  planets 8.807  i-  .007      20 

From  the  lunar  etfuation  in  the  motion 

of  the  Fart h 8.825^.030        1 

From  measurements  of  the  distance  of 

Venus  from  the  Sun's  center  during 

transits 8.857  i  .023        2 

The  mean  errors  which  follow  each  value  are  those  which, 
from  a  study  of  the  tleterinination,  it  seemed  likely  might 
attect  them,  no  allowance  being  made  for  mere  possibility  of 
systematic  error.  The  weights  assigned  are  convenient  snuill 
integers,  generally  sticli  as  to  make  the  weight  unity  corre- 
spond to  the  mean  error  i  0".30,  allowance  being  made,  how- 


158 


THE  80LAR   PARALLAX. 


?J 


ever,  for  doubt  as  to  what  value  should  be  assigned  to  the 
incau  error  and  for  the  difl'ereut  liabilities  to  systeiuatit*  error. 
The  mean  result  is — 


[:! 


From  all  deterniinatious;  tt  —  8.707 
Omitting  the  first  result;  tt  =  .s.800  J:  .0038 

The  last  value  ditt'era  from  the  preliminary  value  8".802  of 
Chapter  V,  from  a  change  in  the  weights.  It  will  be  seen 
that  the  different  values  are  all  as  accordant  as  could  be 
expected,  with  the  exception  of  the  two  extreme  ones.  In  the 
largest  value  we  have  a  case  the  principles  involved  in  which 
have  been  discussed  in  Chapter  IV. 

We  can  not  suppose  the  parallax  to  bo  materially  greater 
than  H"..SOO,  and  may  take  it  as  probably  less  than  this.  Thus 
the  absolute  error  of  the  results  of  measures  of  Venus  on  the 
face  of  the  Sun  uuiy  be  considered  as  about  0".0(>  or  0".07, 
which  is  four  times  the  computed  probable  error.  The  prob- 
ability against  this,  eveu  in  the  case  of  one  result  out  of  eight 
or  nine,  is  so  suuiU  that  we  must  either  regard  the  method  as 
being  affecte<l  by  some  systenuitic  error,  or  as  aff'e«!ted  by 
an  objective  probable  error  larger  than  that  assigned.  It 
seems  to  me  the  latter  view  is  not  untenable,  in  view  of  the 
very  wide  range  of  the  possibilities  of  error  which  might  affect 
a  series  of  observations  with  a  heliometer  exposed  to  the  Suu'a 
rays  during  a  period  limited  to  a  few  hours. 

Again,  in  the  photographic  measures,  the  value  of  a  second 
of  arc  in  length  on  the  photographic  piate  enters  as  a  some- 
what uncertain  element.  In  this  connection  it  is  to  be 
reinarked  that  the  measures  of  position  angle  on  the  photo- 
graphic idates,  which  are  not  attected  with  this  uncertainty, 
although  their  probable  error  is  <)uite  considerable,  give  a 
value  of  the  solar  parallax  much  smaller  than  the  measures  of 
distance. 

Much  more  embarrassing  is  the  value  which  results  from  the 
mass  of  the  Earth.  We  here  meet  in  another  aspect  the  same 
deviation  which  we  encountered  in  determining  the  mass  of 
the  Earth  from  the  secular  variations,  and  on  which  we  post- 
poned a  couclusion  (§G4).     This   determination  rests  very 


• 

! 


TO,  80] 


MOTION  OF  THE  KoDE  OF  VENUS. 


109 


largely  on  tlu*  inution  of  tlie  imhIo  of  Venus,  as  (Ictormined 
from  the  transits  of  1H»1  and  17«»5>,  It  i.s  tnu'  that  results  of 
meridian  observations  are  combined  witli  them;  but  no  cxpla- 
nation  is  thus  atlorded  of  th(^  ditlleulty,  bei'ause  the  results  of 
these  observations  agree  with  those  of  the  transits  (r.  §3J>). 
What  adds  to  the  embarrassment  and  prevents  us  fnun  whitlly 
discarding  the  suspicion  that  some  disturbing  cause  has  acted 
on  the  motion  of  Venus,  or  tiiat  sonu-  theoretical  error  has 
crept  into  the  work,  is  that,  of  all  the  determinations  of  the 
solar  ]>arallax  this  is  the  one  which  seems  the  nu)st  free  from 
doubt  arising  from  possible  undiscovered  sources  of  error.  It 
is,  as  we  shall  i»resently  see,  really  entitled  to  twice  the  relative 
weight  assigned  it.  As,  however,  the  determination  rests 
mainly  on  the  motion  of  the  node  of  Venus,  and  this  again 
mainly  rests  on  the  observations  of  the  older  transits,  I  have 
made  a  reexamination  of  the  results  of  these  transits  with  a 
view  of  reaching  a  nunc  e\a«'t  estinuite  of  the  sources  «»f  error 
and  the  nuignitude  of  the  mean  error.  In  this  reexamination 
I  have  regarded  the  Sun^s  parallax  as  a  known  (puuitity  e(|ual 
to  8".71>8,  aixl  then  obtained  the  results  of  the  ol<l  observations 
of  the  transits  on  the  suppositi(»n  that  the  only  (juantities  to 
be  determined  were  the  correcti<»ns  to  the  relative  heliocentric 
positions  of  Venus  and  the  I'2arth. 

RedisvH8Hion  of  the  motion  of  the  node  of  Veniis. 

80.  In  discussing  the  observations  of  17<»1  and  17(i'.>  (.l«^ro- 
nomieal  Paperti,  Vol.  II,  Part  V),  I  introduced  a  quantity 
expressive  of  the  error  in  the  observed  time  of  contact  arising 
from  imperfections  of  the  telescope  aiul  atmospheric  absorp- 
tion and  dispersion.  The  constants  on  which  these  errors 
depend  are  represented  by  synd)ol8  kt  and  A:>  As  1  have 
worked  up  the  observations,  the  ultimate  result  of  each 
observation  of  contact  is  the  value  of  an  unknown  <iuantity, 
3c,  which,  were  there  no  imperfections  of  vision  and  were  the 
radii  of  the  Sun  and  Vemis  accurately  known,  would  rei)reseut 
the  correction  to  the  tabular  distance  of  <*enters.  As  a  matter 
of  fact,  however,  we  are  to  consider  6  c  as  e<iual  to  this  correc- 
tion increased  by  a  rather  complex  combination  of  quantities 
depending  ou  the  errors  of  the  assumed  semidiameters  of 


160 


MOTION  OF  THE  NODE  OF  VENTS. 


m 


Venus  and  rlic  8nn,  nn<l  the  tlii(;kncHH  of  tin*  t)iron(l  of  light 
wluMi  it  lirst  bcTunie  visihlf  at  serond  fonfurt,  or  viinisIuMl  at 
third  contiU't.  The  obNeivutions  must  be  so  ('oin))ined  as  to 
eliminate  these  «|uantities.  VVhiit  I  havi>  done  is  to  n-prrsent 
the  undiscoverable  minute  corre<*tioii  to  tU'  thus  arising  by 
the  8ynd)oI  c^  for  second  contact,  and  ^i  for  thinl  contact.  In 
the  present  re  examination  the  absolute  terms  are  reduced  to 
the  parallax  H".7!>H  by  putting  Sn^  -  -  ".05  and  n' =  -  ".025 
in  the  linal  -  nuationsof  the  origiind  paper.  After  each  result 
is  given  the  mean  error  with  which  it  is  aiVected,  as  deter 
ndned  by  the  investigation  in  question.  When  thus  treated, 
the  equations  which  1  have  given  on  pages  .iiH-.'JOH  of  the 
paper  referred  to  give  the  following  muinal  equations  for  rfc, 
the  indeterminates  hi  ami  A*;,  being  retained  as  such  in  order  to 
show  their  tinal  elVect  t)U  the  result. 


1701.     II;     H.5   (Sv 
III;  41.7  6c 

1769.     11;  41.8  6e 
HI;  12.1  (h- 


+ 

0.76  - 

18. 

1  k. 

1 

0.78 

— 

2.81  - 

10. 

2A-3 

:|- 

1.30 

— 

S.OO  - 

104.1  A-, 

± 

1.05 

+ 

o.;m  - 

It 

Au 

1 

0.70 

In  order  to  vary  the  i>roceeding  as  nuich  as  possible  from 
that  of  the  former  investigation,  I  now  express  rfc  in  terms  of 
S\  and  Sfi,  which,  for  the  time  being,  I  take  as  the  cju-rections 
to  the  heliocentric  longitude  and  latitude  of  Venus  referred 
to  the  Karth,  and  these  again  in  terms  of  6r  and  sin  i6f), 
which  latter,  for  brevity,  I  call  u.  The  lirst  transfonnation  is 
made  with  the  coetlicients  of  \).  71,  where  we  have  put  .r  and 
—  y  for  S\  and  rf/J^,  and  the  last  by  the  «'<|uations 


// 


rfA  =  rfr  -I-  0.06  i( 
6(i  =   u  —  0.06  y 

Putting  Ml  for  the  value  of  n  in  1765,  we  have,  in  coD8e(j[uence 
of  the  known  change  in  the  motion  of  the  node, 


// 


In  1761 ;  tt  =  «,  -f  0.11 
In  1769;  m  =  m,  -  O.U 


W|  MOTION  OF  THE  NODE  OF  VENUS.  101 

\V>  tlms  have  the  four  or|nntion^  whhh  follow  for  «l(>teriniiiiii<; 
fVr  iiiul  III,  the  forim  r  hiiiiy  supposed  tlie  same  at  the  times  of 
the  two  transits. 

//  // 

_  ,84  ,Jr  _  .55  »/,  -)-  :^i  =s  +  0.15  —  L»,L'  /...   |   O.m 

+  .73  ._  .m  4-  -.  =  4.  HM  -  0.5  k-.  I  ().(>;{ 
_  .«iO  +  .T.J  +  :,  =  -  0.10  -  L».;{  A  .  I  (».(M 
+  .81         4.  .00       -I-  c,  =  -I-  0.10  —  l.;{  /,-,   I    (MMi 

Kliniituitiiijr      an<l  c,  l>y  siihtiaetiii};  tlie  first  ('<iuatlon  from 
the  thiKl,  and  the  seeond  from  the  fcMirtli,  we  liave — 

.!.■■>  (ir  +  1.28  «,  =  _  iK'jr*  -  0.1  A.   ,    0.10 
.OS  ,Sr  +  1.20  M,  =  +  0.00  -  0.8  A^  ±  0.07 

We  thus  have  hu-  «;  the  vahie 

Wi  =  _  0'  .04  -  0.08  ,)r  -  0.o;i  A-,  -  0.;{0  A,  I   ()".05 

rfpcan  not  be  determined  independtMitly  of  c.  an*'  *.  Assum- 
ing these  quantities  to  be  equal,  wo  have  already  fouml  it  to 
be  uidy  O'.ol'.  and  may  therefore,  to  detmnine  its  luobable 
etfeet  up<»n  the  result  by  assi|,'nin{>-  to  it  the  value 

'h'  =  0".(K>  I   0".22 

In  tii'3  former  paper  i  have  found  for  A.  and  A;,  the  values 

//  // 

J(i  =  +  0.040   I    0,040 

A;,  =  _  0.034  :L  0.040 

A  preliminary  correction  of  -f  2".02  having  been  applied  to 
the  tabular  fubital  latitude,  we  have,  for  tlie  epoch  1705.5 

sin  ifU*  =  4.  l".00  I:  0".00 

Combininj'-  this  result  with  that  of  the  transits  of  IS74  and 
1882,  we  have  the  following  results,  whi<h  are  compan'd  with 
those  of  nu'ridian  observations : 

Transits  of  Venus  alone sin  /  I)tfy^= —2.82 

Meridian  observations  alone      ....  "               _  2.45 

Combined  solution u               — "  71 

Adjusted  with  other  results  (§40)  .     .     .  «               —2.73 

Adopted       ((              —2  77 

6090  N  ALM 11 


102 


MOTION  OF  THE  NODE  OF  VENUS. 


[80 


TIh' mlojiti'd  rt'siilt  is  the  oiuMvliich  si'ciiis  tlio  most  inobabli'. 
For  the  liiial  ])i-<)haliU'  nror  we  arc  to  jiicIikU'  that  ol'  the  pre- 
cesHioii  anil  ol'  tlir  Sun's  longitudes  at  the  two  (>|Mi(hs.  We 
nuiy  estimate  tin*  combinetl  value  of  these  at  I  1",  eoirespond- 
injf  to  an  error  »»r  (»".(«» in  sin  / 1>,  tin.    Tlins  we  have 

sill  /  I  >,()■«=  _  L>".77  I  ".0.S4 

I  coneeive  this  mean  error  to  he  as  real  as  any  that  can  be 
determined  in  aHtiiuiomy.  This  eonvietinn  rests  upon  the  tact 
(1)  that  the  systen»ati<'  eiiors  alVeetin^'  the  lour  eontaets  are 
shown  ti»  be  small  by  the  f^eneral  minuteness  of  the  fotir  values 
of  fVr;  (li)  that  whatever  systematic  errors  may  alVe<t  the 
formation  tu*  disappearance  of  the  thread  of  light  are  almost 
completely  eliminated  from  the  mean  of  the  transits  of  1701 
and  1  "<»".►  by  the  method  in  which  the  observations  have  been 
c'ond)incd.  The  accordaiu'*'  of  the  observations  of  exterind 
conta«'t  nuuli  at  the  sap  «•  i::',»'«'ts  strengthens  this  view. 

The  e<|uat on  thn-  tleiivcd  tak«'s  the  place  of  the  sixth 
e(pnitiou  of  •si.'"'  and  should  have  twi«*e  the  weight  there 
assigned.  As  the  mass  of  the  llaith  determined  by  the  secu- 
lar variations  rests  uuiinly  <mi  this  eipiation,  I  shall  lirst  con- 
aider  it  alone.  Expressing  the  theoretical  secular  variation  of 
sin  trf^  in  terms  oi'  the  above  observed  value,  we  iind  that  the 
observed  motion  (tf  the  node  of  Venus  gives  the  equation 

•   O'MM;  I'  —  LM»".2  I''  —  4'.i":2  y"  =  +0-".48  i  <>".(>.S4  (a) 

which  gives  for  r"  the  value 

y"  =  -  (M)lll  +  O.tMMJ  r  -  0.<»7«i  )'  rl    .nOU» 

The  value  of  the  sidar  i)arallux  for  i"  =  0  is  8"..sil.  Hence, 
for  the  value  expressed  in  terms  of  the  coire<'tions  to  the 
assumed  masses  of  Venus  and  Mercury,  this  equation  gives 

n  =  S".778  +  0".020  r  -  1".08  i-' 
We  have  found  from  the  periodic  perturbatious 

J  =  -  0.055    i  .25 
r'  =  +  0.<M)80  i  .(X)25 


• 

• 

») 

UOLAU  PAUALLAX. 

1U3 

^Vht'iu'C, 

y"  =  -O.OH58  1. 
rr  =      M.TOl'     1 

.(X»80 

This  rcHU 

I  4iiii  not 

1                     that  \\vrv 

siipi 

assi^ 

4)hs( 

,iumI 

rvatioii.  errors  and  nnknown 
to  hv  aflrrttMl  by  any  otluT  iin 

action  H 
•an  »'rroi 

ashlo, 
-  tlian 

We  inive  now  to  j-onHider  liow  far  this  n-snlt  may  l»«'  leron 
cih'd  witii  tlie  «»tlu'r8  by  ciianfjes  in  tlie  masses  of  MiTcni-y 
un<l  N'enus.  Notulinissible  eiuinp:e  in  tlie  t'ornier  rouhl  greatly 
atl'ect  tilt  "eHnlt.  The  «|uestioii  then  arises  wlietlier  tlie  «lis- 
crepaiiey  nuiy  not  be  due  to  an  error  in  the  eoniln(U>(i  mass 
of  Venus.  In  making'  so  lar};e  a  ehan;;e  in  this  elenn-nt,  wo 
iue<'t  with  insuperable  ditlieiilties.  The  observed  motion  of 
the  t'cliptie,  which  is  a  fairly  well-determined  quantity,  indi- 
cat«'s  a  still  further  increase  of  this  mass.  We  may  put  this 
ditliculty  in  another  form.  The  observed  motion  of  the  nrxlo 
of  \'enus  is  a  relative  one,  consistinj;  in  the  conduned  etV«'ct  of 
the  motion  of  the  e(diptic  around  an  axis  at  right  unifies  to  the 
node  of  N'enus,  and  an  abstdute  motion  of  the  orbit  of  N'enus 
arouiul  nearly  the  same  axis.  This  motion  ol'  the  ecliptic 
depends  mainly  on  the  mass  of  Venus;  the  absolute  motion 
of  the  orbit  of  Venus  mainly  on  that  of  the  Karth.  If,  now,  we 
determine  the  motion  of  the  ecliptic  from  observation,  we  shall 
find  that  the  relative  motion  of  the  orbit  of  Venus  still  unac- 
counted for  is  yet  ;;reater  than  we  have  supposed  it  to  be,  and 
shoidd  tluM'efore  llnd  a  yet  smaller  mass  of  the  IC;irth  than  that 
heretofore  concluded. 

The  deternunation  of  the  mass  of  Venus  already  made  from 
observati(»ns  of  the  Sun  and  Mercury  seems  to  adndt  of  no 
doubt.  We  can  not  coiic«'ive  that  the  mean  of  liftecn  deter- 
minations, nnide  duriuju;  one  hundred  ami  thirty  years,  at  dif- 
ferent observattuies,  which  tleterminations  are  so  separated  as 
to  be  entirely  independent  of  each  (»ther,  can  be  atfected  by 
any  considerable  common  error.  The  entire  acccudance  of  the 
result  thus  reached  from  the  i>eriodie  i)erturbations  produced 
by  Venus  with  that  from  a  combination  of  all  the  secular 
variations,  as  shown  in  Chapter  VI,  strengthens  the  result 
yet  further.     Unknowu  actions  and  i)ossible  defects  of  theory 


m 


■  1 

i 


104 


SYSTEM ATir   EUU«>RS  OF   PARALLAX. 


•O.M 


asi)|«'.  it  st'eins  to  iii«'  tliat  tin-  valiu'  of  tlu*  solar  parallax 
ilcrivi'd  troin  tins  discussion  is  h-ss  M]H'n  to  doiibt  from  any 
known  ciuisi*  than  anv  (lotcrtnitnitioii  that  can  be  nnnli*. 


I'nxsihlr  si/sh'tnotir  nvnys  in  (IctcntkiuatioHH  n/  Ihr  panilhi.r, 

SI.  Wo  have  now  to  return  to  the  otln'j-  values,  in  order  to 
see  to  what  i^xtent  they  may  be  afl'eetejl  by  systematic  error. 
I  have  aheady  excused  myself  from  discnssiny  the  validity  of 
the  assumed  relation  between  the  c<tnstant  of  aberration  and 
the  velocity  of  li<«ht.  because  tlieie  is  nothin<i'  valuable  to  be 
said  on  the  subject,  and  have  alluded  to  the  possible  sour/es 
of  systematic  error  in  the  I'ulkowa  determinations  of  abei-ra- 
tion.  It  is  worthy  of  attt'iition  here  that  the  very  best  of  these 
determinations,  that  of  Nvkkn  with  the  prime  vertical  transit, 
in  respect  to  tlic  care  w  ith  which  it  was  imide.  and  the  jreneral 
accordance  of  the  entire  work  throuj,'hout.  aWi'n  a  residt  most 
accordant  with  that  under  consideration.  In  fact,  to  the  \  alue 
S".77  of  tin'  solai'  parallax  coii«'s]K»nds  the  value  LM)"..").")  of 
the  constant  of  aberration,  which  is  larjit-r  by  only  «>".(>'-•  than 
the  result  of  Nykkn's  best  dj-terndmitions. 

fVs  for  niiscellane«>us  determinations  of  the  constant,  it  is  to 
be  remenil>ered  that  the  corre(!tions  api)lied  to  a  part  of  the 
separate  values  on  a<'«'ount  of  the  ('InnuUeriau  inequality  <»f 
latitude  arc  som<'what  doubtfid,  and  the  ^'en<'ral  mean  uuiv 
have  been  affect«'«l  by  a  few  humlredths  of  a  se«'ond  in  conse- 
queuce.  It  is  not,  however,  possible  to  determine  the  amount 
of  the  (orrection,  I'xcept  by  an  exhaustive  rediscussion  of  the 
whole  of  the  orifjinal  observations,  ami  even  then  the  result 
wouhl  still  be  doubtful. 

J'ext  in  the  order  of  weight  we  ha>e  the  lesnlts  of  measures 
oc.  the  nunor  planets  with  the  heiiometer,  on  (Jin/s  plan.  I 
!iave  already  remarked  upon  the  possible  error  in  such  obser- 
vations aiisin^i  from  the  probal>h>  ditterence  of  c(dor  between 
the  platiet  and  the  star.  A  hypothetical  estinuite  of  the 
am«nint  «>f  this  erroi'  is  worth  attemptinj;.  Let  us  assnnu'  that 
in  the  case  of  a  minor  planet  the  mean  of  the  visdde  spec- 
trum corresponds  to  tlu-  line  1).  and  that  in  the  ca^e  of  a  star 
tKe  same  mean  is  halfway  between  the  lines  1)  and  K. 


811 


SYSTEMATIC   EUBOBS  OF  I'AUALLAX. 


1(>5 


The  imlcx  of  it'Ciiutioii  of  air  has  Immmi  (letrniiined  iiule 
]H>ii(1eiitIy  by  Kin  tlkr  aixl  Lokkmz  for  the  dirt'eieiit  rays. 
Tlu'  mean  of  their  results  (or  the  ravs  I>  and  K  is 


1m »r  I),  n  =  l.(M»OLM >:.>«> 
For  K.   )i  =  1.(100  IMMO 

These  results  are  accordant  in  ;,'i\iiiji  a  dispersion  between 
these  two  lines  equal  to  about  .(KKST  of  the  total  refra<'tion. 
We  have  hypothetically  taken  the  extreme  possible  ditVerence 
between  ])lanet  and  star  to  be  ohe-iialCof  this.  At  an  altitmU^ 
of  l."*"^,  where  the  refraction  is  about  (10",  (lie  err()r  would  be 
0".1I.  At  iin  altitmle  of  IW  (he  error  would  be  ((".I'O.  We 
are  thus  led  tt»  the  noteworthy  conclusion: 

ll'thi'  tli()'nin(«'  hi  tinrii  i'l'  .^fHrtra  <>/  <i  tiilnor  jthiint  iinil  n 
I'oiiifKirinon  sttir  is  siirh  flint  ihc  nniuis  of  ihiir  rispivni'c  riHihlc 
fijH'ctrii,  or  tlif  tipfHirnif  oinninits  vf  tin  ir  nspwtlre  rvf'rnrtinuH, 
(1i()'<r  hji  oiii  triitli  nf  till'  sfHirr  hilirnii  />  innl  11.  iin  ivror  of 
0".(>S  or  (>".(>■' 1  111(11/ he  proihinit  in  tJir  iijiitnrciit  paraHn.r  of  tlw 
phiiu't. 

The  (pu'stion  tlips  urisiii^' may  be  i-eadily  settled  by  measures 
with  the  hcliometer.  The  distances  of  pairs  of  stars  ditVcrin;; 
as  widely  as  possible  in  ccdoi-  sliould  be  measured  at  ditVerent 
altituiies,  when  one  is  nearly  al>t>ve  oi-  bdow  the  other,  in 
order  to  see  what  ditlerence  of  refraction  depcndin;^  on  the 
color  is  iiulicated.  A  colored  doultle  star,  such  as  fi  (.'ygni, 
miji'ht  also  be  used  for  the  same  purpose. 

The  minor  planets  are  oi'  (litfer«Mit  <'olors.  I  am  not  aware 
of  any  evidence  that  \'i<'t(uia  or  Sapplm  <lilV»'r  in  color  IVom 
the  av«':a}ie  of  (he  stars,  but  1  bebevi'  tha(  Iris  is  somewhat 
yellow,  or  red<lish.  Now,  in  tiiis  connection,  it  In  a  siyriilicant 
fact  that  th"  parallax  found  from  ol)ser\  ations  of  Iris,  S'  .,Si»,"». 
is  the  laryest  by  (ii(,l/s  methoil. 

1  hav«>  already  remarked  tliat  the  valin*  oi'  tiie  s<»lar  parallax 
derived  from  the  paralla<-tic  etpnition  oftlieMoon  is  one  of 
which  the  probable  mean  ei  ror  is  subject  to  imcertainty. 
While  it  is  true  that  the  value  may  be  smaller  than  that  we 
have  assigned,  we  nnist  also  admit  that  it  may  1h>  much  larjjer. 

The  probable  error  of  the  determinati<Mi  by  the  lunar  equa- 
tion of  the  Karth  is  laryj'r  than  (hat  of  any  other  method.     At 


166  RESULTS  FOR  THE   SOLAR   I'ARALLAX.  [82 

the  Riimo  time  1  do  not  think  that  it  is  liable  to  systematic 
error,  and  we  must  therefore  rejjard  the  mean  error  assigned 
as  real. 

RisuHh  /of  the  solar  luirallax  after  mnkiuj/  alhnranci'  for  pmh- 

able  HyHtvmuth-  vrrorn. 

.H2.  Let  us  now  st'c  wlu'tlu'r  \\v  can  rrach  a  safisfartory 
result  by  making'  a  liberal  allowance  lor  the  nioi'(>  or  lesH 
probable  sources  of  systematic  error  Just  pointed  out.  The 
nnxlilications  we  maU(^  in  the  weights  lornu'rly  assigned  are 
these:  We  redu<'e  the  weight  of  (llLJ.'.s  Ascension  result  to 
oiH'  half,  owing  to  the  unceitainty  arising  from  the  color  of  the 
planet  Mars.  We  r«'tain  the  iMilkowa  <leterminatu»ns  of  the 
constant  of  aberration  with  tln-ir  full  weight,  but  re«luce  the 
weight  of  the  miscellaneous  di'terminations.  In  the  case  of 
the  parallactic  ineiimility.  we  rc«lnce  the  weight  for  the  reasons 
already  gi\  «'n.  We  omit  Iris  from  the  deteriiiination  from  the 
minm*  planets.  We  also  reduce  to  t»ne  half  its  former  value 
the  relative  weight  assigned  to  measures  of  \'«'nus(»n  the  Sun, 
on  the  tli(>ory  that  the  actual  nn'an  error  nnist  be  larger  than 
that  given  by  the  disi-oidanci'  of  results.  Our  combination 
will  then  In-  as  Ibllows: 

From  the  iiKidoii  of  thf  uoile  of  \'riiiis          ,     .     .    n-sf  8.7<»H  10 

Frotn  iiiiA.'fi  Asccnuioii  olmcrratioHs      ....  S.T.SU  I 

From  thr  Pidkoirti  ionslaiif  of  alnrr(ttioit  .     .     .  S.7!K{  40 

From  vontortH  of  Vnnis  iritli  llir  tSiiii's  limh     .     .  S.TiU  ."{ 
From   li(liomr(tr   olmcrraiions  on    Victoria    ami 

Sapi»lii> H.71M>  .') 

From  thr  jtarallartir  iiminalitji  if  tlir  Moon    .     .  H.TIM  10 
From  niiHfi  ilannms    ilrtvrminationx   of  tlir    con- 
stant of  ahrrration     S.SOtJ  10 

From  the  hntar  incfinalitii  in   the  motion  of  the 

Forth H.818  1 

From  nicannres  on   Venus  in  transit S.H.*)?  1 

Mt-an  residt,  ignoring  the  llrHt;  8".70«'i."»  I  .004.") 

This  mean  result  still  ditVers  t'rouj  that  given  by  the  motion 
of  the  node  <  f  \enus  by  nnue  than  ii\e  time.s  the  i»robable 
eri*«rof  the  litter,  and  is  vet  farther  from  the  combined  result 


821 


llESULTH  FOR  THE  SOLAR  PARALLAX. 


107 


of  all  the  st'culiir  variations,  so  that  no  nM-onciliation  is  broufjht 
about. 

The  eiiiharrassin^r  (pu'stion  whith  now  morts  us  is  whether 
we  have  here  sonx'  uni\iio'.vn  einise  of  tlilfereiire,  <»r  whether 
the  (liscrepaney  arises  from  iin  aceiileiitai  aectiniulatioii  of 
I'ortnitous  <'rrors  in  tht;  separati*  <letrriniiiatioiis.  VVe  iuive 
ah'eatly  discusse*!  tiie  former  hypothesis,  ai..l  liave  been  unable 
to  find  any  reasonably  i)robabh;  eans«>  of  al>norinal  action. 
The  motion  of  the  phmes  of  the  orbits  is  tluit  whieh  is  h'ast 
bk«'ly  to  (h'viate  from  th«'ory,  because  it  is  independent  of 
all  forms  of  action  dependin^^  u])on  «listanee  from  the  8nn, 
or  upon  the  vlocity  of  t lie  i)lanct. 

An  examination  and  comparison  of  all  the  results  shows  one 
curious  feature:  the  unanimity  with  which  the  secular  varia- 
tions speak  auainst  the  larye  value  of  the  solar  parallax,  or 
of  the  mass  of  the  llartli.  as  tli  *  one  quantity  at  fault.  The 
adopted  motion  of  tiie  node  of  Veinis  is  sustained  not  only  by 
the  meridian  observations,  but  l>y  th«'  external  contacts  at  the 
transits  <»f  17«il  and  17(»'.»,  and,  wiakly,  by  a  comparison  ol  the 
transits  of  1S7I  and  ISSi*. 

if  we  deteriiiiiH'  the  coireclion  of  the  inassoftlie  Marth  from 
other  se<-nlar  variations  than  that  of  the  node  of  Venus,  by 
the  e<|uations  of  ^  (i.'l.  we  havi',  alter  eliminating  tlu'  masses  of 
Mercury  und  Venus, 

r"  =  —  n.02!»;    p.  V.    I    .(US 

If,  insteivl  of  eliminating  tiiese  values,  we  put 

I'  —  +  .08  J    U'  ss  +  .0080; 

we  have 

I'"  =  -  n.OLMi:  p.  e.   i   .011 

In  eaeh  ca«e  the  value  of  tlu'  parallax  is  yet  smaller  than  that 
found  from  the  motion  of  the  node  of  N'enus.  1  have  already 
remarked  that  the  observed  motion  of  the  ecliptic  indicrates 
an  iii'-rease  of  the  nniss  of  Venus. 

The  (piestion  thus  lakes  the  form,  whether  it  is  possible  that 
the  mean  of  the  ^even  determinations  of  the  Nolar  [tarallax 

t 

TT  =s.S".707  I    "Am') 


ir,a 


DEFINITIVE  ADJUST  «ENT. 


[82,  83 


can  with  reasonublc  possiljility  be  in  error  by  an  amount  the 
wjrreetiun  of  wliieli  would  brin^'  it  within  the  ranj;e  of  adjuHt- 
inent  of  the  other  (|nantities. 

From  what  has  aheady  been  said  of  the  systenmtii;  errors 
to  wiiieh  tnery  one  ()f  the  determinations  may  be  Habh',  it  is 
eviih'nt  that  we  shouhl  liave  no  dir.ieulty  in  aeeepting  the 
iieeessiiiy  reduction  «if  v.w.h  of  tln^  separate  values.  The 
improbability  which  meets  us  is  uot  so  nuich  the  anumnt  of 
the  individual  errors  of  the  <leterminations  as  the  fact  that 
seven  of  the  eif>ht  independent  deterinimitions  should  all  be 
hiifj^ely  in  error  in  the  same  direi'lion.*  Still,  under  the  eir- 
eumstaiu'«'s,  we  must  admit  this  jjossibility,  and  nuike  what 
seems  to  be  the  best  adjustment  of  all  t!:f'  '"esults. 


V- 


Ih' fin  it  ire  (nljustmnit. 

8.S.  In  niiikintj  the  delinitive  adjustment  1  shall  i»roceed  on 
the  supposition  that  no  correction  is  necessary  to  theadoptetl 
mass  of  Mars,  I  also  {^o  on  the  i>rinciple  that  in>  result  is  to 
be  rejeete*!  on  account  of  «loubt  or  discordance,  except  when 
it  is  alfcctcd  witli  a  wcirestalilished  causeof  syst«'matic  error, 
and  sIkiws  a  larye  deviation  in  tlu!  direction  in  which  thii 
cause  would  act.  At  the  sanu'  time  it  will  be  admissible  to 
diminish  the  weiylnts  in  special  <ases,  on  account  of  causes  of 
systematic  eiror  wliic^h  we  know  to  exist,  althouj^h  we  can  not 
determine  the  <lirections  in  which  they  woulil  act;  and  also  oi, 
account  of  d»niations  so  wide  as  to  show  that  the  probal»le 
error  of  the  >esult  must  have  been  greatly  underestimated, 
l^roceediiiji'  on  this  plan,  we  mi^ht  rt'weijiht  the  last  eij;ht 
results  fui'  the  snlar  parallax,  so  as  to  j^et  a  result  slijihtly 
dilVerent  from  S'  .7!»7.  Ibit  1  <loubt  whether  sncli  a  reweight- 
iu'fi  would  not  involve  an  objectionable  bias. 

VV«'  mijihl  diminish  the  weight  of  tin'  result  given  by  the 
iMilUowa  constant  of  ikb(>rration  on  the  gi'onnd  that  no  one 
nnthod  ;  hould  hav«'  so  luepoiulerating  a  weight  as  this  has. 
li  we  did  so  the   result   might  be  increased  to  .S".8(K».     Wo 


For  a  vi-r\  McartlmiK  criticisin  nt'tlio  syHli'iiiatir  i-rrnrHwitli  which  the 


'ffd,  n'foi— luc  may  1 


*e 


(Iftt'i'iiiinatioiiH  <>t'  thi>  solar  parallax  may  nt 

iiiatlc  t(»  the  liiHt  two  arti«  Kh  1»,v  Dr.  David  Giu.,  iu  WA.  I  of  The  obterva' 

'"'■.'/■ 


831 


DEFINITIVK  AD.I I  STMKNT. 


Ui\t 


m'gbt  very  larffoly  increase  the  relative  weiylit  assignetl  to 
the  helioineter  observations  on  N'ietoria  and  Sapplut.  but  no 
admissible  in«-reast'  would  appreciably  ehan;ie  the  result.  We 
niigiit  iflso  diminish  the  relative  weight  of  the  largely  dis 
conlant  result  derived  from  nn>asures  of  Venus  «luring  tninsii. 
lint  as,  by  throwing  out  this  result  altogether,  we  shoidd  only 
diminish  the  mean  by  ".«KH.  it  is  seareely  worth  while  to  do 
so.  -Vltogether  no  rediseussion  (»f  the  relative  weights  seems 
necessary. 

On  the  other  hand,  the  weight  whi<'h  we  assign  to  the  mean 
result  will  enter  as  a  very  impoitant  factor  into  tiie  final 
adJiiStni«Mit.  This  is  a  point  on  which  ii  is  impossible  to  reach 
a  positive  numerical  conclusion  by  any  mathematical  process. 

If,  as  one  extreme  case,  we  considi'r  that  the  mean  error  ot" 
eacii  separate  result  Ci»rres[«)nds  to  I  (»".03  for  weight  unity, 
we  shall  have  a  imnin  error  of  I  ".(KKJo  for  tlie  value  s".7J»7. 
The  residt  will  not  be  very  dillerent  if  we  determine  the  mean 
error  from  the  discord'ince  of  tin'  eight  sepaiate  results.  On 
the  other  hand,  if  we  include  the  deviation  of  tin*  result  givi-n 
by  the  motion  of  the  node  of  N'enus.  the  uhmii  erriU"  for  weight 
unity  will  be  increased  to  L  0  '.(MM5,  The  latter  is  undonbt 
edly  the  most  logical  c<»urs(>,  so  long  as  we  proceed  on  the 
hyp«)thesis  that  the  d«'\iations  of  the  tinal  adjustment  <'au  all 
be  exi»lained  as  due  to  fortuitous  errors.  If  we  include  a  cotn 
l)arison  with  the  results  of  all  the  secular  variations  we  shall 
have  a  yet  larger  mean  error.  To  show  the  lesult  of  assigning 
one  weight  or  the  other  I  siiall  make  tw(»  solutions,  A  and  11, 
in  one  of  which  a  h'ss  and  in  the  otiier  a  gieater  weight  will 
be  assigned. 

To  the  value  S".7!>7  i  .0(»:»  or  j  .007  of  I  lie  solai  parallax 
correspond.'. 


]'"  =     -  (MH!»    i     .(MHI'm.!' 


.(10". 


According  as  \\v  assign  one  weight  ur  ihe  other  to  this  rcMilt, 
we  may  take  as  the  corresponding  ei|iialion  ol  condillon  )f 
weight  unity 


or 


(A); 
(B); 


4()(h  "  =  -  2.0 
000-"  =  -  2.U 


(") 


170 


DEFINITIVE  ADJISTMENT. 


183 


The  masses  of  Venus  and  Mercury,  ileterinined  by  niethoils 
in(le)>(U(lently  of  the  seeuhir  Viiriat  ions,  also  enter  as  conditions 
into  the  adjustment.  I  have,  however,  made  a  revision  of  the 
preliminary  adjustment  ^^iven  in  §  <»4,  the  hitter  heiiifif  l{ase<l  on 
tlie  results  of  §v^  .i^-.'^S;  whereas  it  is  better  to  use  the  detini- 
tive  restdts  of  the  combination  used  in  §  40. 

For  the   mass  of  Mercury  the  result  Ibund  in  ^.W  by  the 
lust  combination  is 


II  0.;{5 
7i>4.MI(K» 


ih) 


The  Viduea  of  the  denonnnator  corresi)on«lin^  to  the  mean 
limits  here  assijrned  are 

.^SIKMMMI  and  12lM(M)0(> 

These  limits  are  tut  wide  as  to  include  all  admissible  results  for 
the  nmss  of  Mercury.  Moreover,  we  can  not  dellnitely  say  that 
the  value  (h)  of  this  mass  is  markcilly  {;reater  or  h'ss  than  Hint 
jjivcn  by  the  wei^ihtccl  mean  of  all  other  results,  since  we 
miyht  so  weight  the  latter  as  to  jfivc  a  result  {jreater  or  less 
without  transcending  tlu'  bounds  of  judicrious  Judf^ment.  I 
conceive,  theretbre,  that  we  are  Justilled  in  reducinj,'  the  mean 
error  to   |  (».l,M»,  which  will  yiv*'  i«s  the  e<|uation  of  cjondition 


and  hence 


(>.().").•>  I  O.L'5 


U).v=  -  ().2'J  II 


{<') 


When,  in  the  normal  «'<|uation  for  the  mass  of  Venus,  given 
by  the  observations  on  Mercury,  we  sulistifut*^  the  values  of 
the  secular  \ariation.s  found  from  the  general  condiination  of 
§  Mi,  the  result  is 

)'=-(MHl4 

Coudiining  this  with  the  result  from  the  Hun,  wi>  have 

1  '=  -U.UllT 

In  view  of  the  (iwt  that  the  nmss  lU'rivcd  ft'oni  observations  of 
Mercury    may  be  alVecte<l  by  systematic  eirors  of  the  kind 


831 


DKFIMTIVE   AD.IISTMENT. 


171 


Hhown  ati<l  (liscussod  in  §'».'{,  flu-  tiM-aii  error  forincrly  assi<riie<l 
to  tliis  result  should  be  soiiiewhiit  diiniiiisliiMl.     The  result  is 


in'  = 


1 
4(Hi  (KK) 


From  this  we  have 

) '  =  +  0.0084  I   .00;{() 
For  the  epilation  of  coixlition  of  Wfijiht  unity  I  take 

3; 50  »'  =  -f  i'.,S 


(d> 


NN'ith  tlu^se  eqiuitions  of  coiidiiion  wr  liave  to  coinbine  the 
eleven  e(|uatioiis  of  «0.i.  wiiich  we  nse  unehaii;xe«l,  exeept  that 
we  double  the  wei;iht  assifjiied  to  tlu'  sixtii  ••((nation,  that 
deiived  from  the  motion  of  tin;  node  of  N'einis,  on  a('<'onnt  of 
the  .smaller  i>n>l»able  eiior  of  tin'  I'esnit  of  our  precedinji;'  redis- 
eussioii,  and  use  the  value  of  the  alisolute  teiiii  fonml  in  ^^SO. 

If  we  aeeept  the  view  that  all  the  perilielia  nn»ve  accordiiij; 
to  the  sauu^  law  of  j^ravitation  towaid  the  Snn,  nannls ,  that 
expressed  by  llAiJ/s  hypothesis,  then  the  \aIneol  the  (pniu- 
tity  '')  in  the  formida  expressiii};  the  law  of  /^navitation  is  so 
well  determine<l  by  the  motions  of  Mcrt-nry  that  it  beconx'S 
legitimate  to  ns»^  the  obser\t'd  nM»tions  of  the  peiihelia  of  the 
other  thi'e«'  jilanets  as  eqnations  of  eoinlitioti  ISiit  sine*'  it  is 
not  impossible  that  the  minor  planets  between  Mars  ami 
Jupiter  may  have  an  appreciable  tntuienee  o\\  the  nn>tionof 
the  perihelion  of  Mars,  it  is  a  <|Uestion  whether  we  should  not 
exclude  that  motion  fiom  the  equations. 

The  conditional  equations  ^'iven  by  the  motions  of  the  thieo 
perihelia  in  question  aie  found  l»y  comparinH"  the  resnitsof 
M4t),  54,  and  (»1.     Tliey  are 


40.r  -I-.     0  /'  +  20  r"  =:  -f  1.0 

14     4-  40      4-0      =  -  0.;; 
li     -  l.l      +  0!        =  4-  (►.: 


{«> 


Tin'  conditional  eqinitions  to  be  combined  aic  the  eleven 
e(|nations  of  i(i;$,  the  sixth  of  which  is  to  have  <louble  weij^ht, 
and  the  six  eipnitions  {a),  (f),  ((/),  ami  (e). 


ha 


172 


DEFINITIVE  AD.irSTMENT. 


[83 


Tlu'  noniial  (>(|ii!itioiis  to  wliirli  \v«»  arc  thuH  UmI  arc  tli« 
i'ollowiiig,  wliicli  sIkiw  tlic  rcsiiltM  of  the  tour  (■oinbiiiatioiis  we 
may  inakt>  accordiiiK  as  we  iihc  (A)  or  (H)  tor  tiic  4M|iiati()ii 
givt'ii  by  tlic  mass  of  tlio  Martli,  and  omit  or  iiirliuh'  tlic  third 
('(|iiiitioii  (f/),  which  18  ;j;ivcii  by  the  motion  of  the  perihelion 
of  Mars. 

{n.)   Iwhiilitiff  the  motiitn  of  tliv  perihelion  of  Marx. 

\)  m-j  -       7  I  17 1'  -     11  .'{.Tm"  =  +  L'LM) 

-  7  117     +207  171      4-H>«7l-'7       =-587 

_  II  ;{.{.-»    4-H»S7*J7      +  MM».'{0(>      =-;{;iSS(A) 

-  1 1 ;{.'{.'»    -I-  KiH  7l'7     -f  (i(Mi :\m     =  -  i.L's  ( \\) 

{fi.)  (hnHtintj  the  motion  of  thr  prrihelion  iff  Mot'ft. 

1Mi(K5.r  -        7  iL'l  I''   -     11   157 1"  =  -f  L'lH 

-  7  121      -fL'(»7  0(»;{       +  HIMoL'O        =-.'.78 

-  11  i:»7    4-  n;u:.LM>    -i-  4(r.»r»78     =  -  'MM  (A) 

-  11   l.'»7      -I-  Hl'jr.L'O       +  ()0L'."i7S        =  _  l;{71  (M) 


The  lesubs  of  tlie  solutions  in  tin-  f'onr  eases  are: 


A  a 
.r     f  0.0117 
I'    +0.117 
!'■   -1-0.001  .11 
,/"  _  (i.no!»  7.{ 
I  J-  „t        C'lKOOOO 
1  -r  in'        lOS  L'.iO 

TT      8".7s;s 


A/i 
4- (MM  12 
+  0.112 
+  (UHll  ti(» 

—  o.(noo."» 
i>  r»(;7  000 

lOS  120 

.H".7H2 


+  0.01(il 

+  o.n»i 
+  0.00:1 10 
—  0.007  70 
0  100  000 
lOH  7.50 
.S".7.S1> 


+  0.015S 

+  o.l."»s 

+  0.00;  {  2."» 
-  0.007  87 
(;J77tMK> 
408  070 

8".78.S 


1  eonceive  that  il'tlie  se<'iilar  vaiiations, especially  the  motion 
ot  the  no«le  of  \cnus,  arc  not  atVceted  l>y  any  unkn(»\vn  cansc, 
some  mean  between  these  shcnild  be  r«';iarded  as  the  most 
])robablc  solntion.  Tiie  resnif  does  not,  howevei'.  biin;;:  alxiut 
a  satisfiictory  leconciliation.  \V«'  still  lind  onrsi'lvrsecnifronled 
by  this  embarrassing  dilemma :  llilher  there  is  .somethinff 
abnormal  in  connection  with  the  node  of  Venns,  due  to  an 
nnkin)wn  canse  aetinj;  on  tlio  planet,  to  some  extraordinary 
errors  in  the  ob.srrvations  or  their  reduction,  or  to  some  error 
in  the  theory  on  which  the  di.senssion  iu  based,  or  the  deter- 


N 


8.'i,  Sl,H.'»|  ADOI'TKI)   I'AUALLAX   AND   MASSKS.  1 7.*t 

minatioiiM  of  tli<>  solar  parallax  an>  lu'arly  all  in  error  in  n\ni 
(lins-tioii  by  aiiioiiiilH  wliicli  are,  in  more  than  one  case,  «|uite 
snrprising. 

VoHHiblv  rauHvH  iff  the  ohs^Tiu  il  ^liMronhnirt'M. 

84.  Two  possible  causes  of  <liseor<lanee  may  he  siig^jested, 
oneot'wiiicli  lias  not  been  tonelied  upon  at  all  in  the  |)reei><lin^ 
chapters,  and  one  perhaps  inadeipiately.  As  to  the  hy|)othesis 
of  lion  spheiieity  of  the  Sun.  considered  in  ».")«»,  it  may  br' 
remarUed  that  Dr.  IIakt/hk  shows  that  an  dliptieity  of  the 
Sun  Hiinicient  to  produce  the  ol)ser\ed  motion  of  the  peiihelion 
of  Mercury  would  eans«'  a  diii'ct  iiKttioii  of  "i".!  in  the  motion 
of  the  node  of  Venus.  This  would  correspond  to  a  ehanjie  of 
{\".'M\  in  tlu'  valiio  siin'I), 'V  and  would  therefore  ;;o  far  toward 
reeonciliny  the  discrepancy.  Ibit  it  is  easy  to  s«'e  that  this 
cause  would  produce  a  secular  motion  of  —-".0  in  the  inclina- 
tion of  Mercury.  W'c  have  seen  that  the  observed  motiim  of 
the  inclinati(Ui  already  cxcei'ds  the  theoreti«al  motion  by  ()"..'{S; 
so  that  iiitrodncin)L>  the  hypotheses  of  ellipticity  of  the  Sun  wo 
should  have  a  discrepancy  of  about  .{".0  between  theory  and 
observati(Ui.  This  iMUidusion  aloin'  seems  fatal  to  the  theory, 
which  otherwise  has  been  shown  to  be  s<'arc«'l\'  tenable. 

The  other  possible  cause  is  an  iiuMpiality  of  Ion;;  period; 
especisilly  one  dcpendinj;  lui  the  ar;.;uineiit  ]'M"  —  Sl'  which 
has  a  period  of  about  two  hundred  and  forty  three  years.  A 
very  simple  <'oiiiputation  shows  that  the  coelllcient  of  this  term 
is  only  of  tiie  order  of  magnitude  O'^Ol. 

It  is  a  curious  coincidi-nce  that  if  we  had  neylecled  to  add 
the  mass  of  the  Moon  to  that  of  the  ICarth,  in  eompntiii;:  the 
.secular  variati«»ns,  the  discrepain-y  would  not  Ini'.e  i'xisted. 

Adopfnl  rahics  of  thv  doubtful  (jKantiti'n, 

8.">.  The  practical  ipiestion  which  has  been  before  the  writer 
in  working  out  tin*  pre<e<lin<f  results  is:  What  values  of  the 
constants  should  be  used  in  the  tables  of  the  celestial  motions 
of  wlii<'h  the  results  of  this  (iisciission  arc  to  form  the  basis? 
Sliouhl  weaiiii  simply  at  ^rettiii;;  the  best  agreement  with  obser- 
vations by  correi;tions  more  or  h'ss  eini)iri«al  to  the  theory? 
It  se«'nis  to  me  very  clear  that  this  (|uestioii  should  b<^  answered 
iu  the  uej^atlve.    No  concluMioiis  couhl  be  drawn  from  future 


174 


AUOI'TKI)    I'AKAI.I-AX    AN1>    MASSKS. 


l» 


(■<)in|tiii-iHoiis  of  such  talilcH  witli  obsci  vittiuiiH,  except  after 
lediiciiijt  the  tahuhii  results  to  some  coiisistiMit  thtMiry.  Tho 
iiii|iosilioii  of  siicli  a  hiWor  upon  the  future  iuvestiuiitoi-  is  not 
t«i  he  tliou^ht  of.  MoreoNcr,  there  is  no  ctMtaiiity  that  tho 
tal»h's  whii'li  wouhl  l>est  represent  past  ohservations  would 
also  i test  lepH'sent  future  ones.  Our  tahlus  must  he  fouudetl 
on  sonu>  perfectly  consistent  th«>ory,  as  simple  as  possible,  the 
t'lcMU'iits  of  which  shall  be  so  chosen  as  best  to  represent  tho 
obs(>rvations. 

In  choosing  the  theory  an<l  its  constants  we  have  nniun  n 
certain   rant;e.     If  we  accept   the  lu'cessity  of  assumini;  the 


secular  variations  of  the  orbits  of  Mercury  and  Venus  to  be 
atfected  by  the  action  of  unknown  nnisscs  of  matter,  then  the 
simplest  course  to  adopt  is  to  ciuistruct  <Mir  theory  on  the  sup> 
position  of  a  planet  or  ^Moup  of  planets  between  Mercury  ami 
Venus. 

It  s(>ems  to  me  that  the  introdiU'tion  of  the  action  of  such  a 
{froup  into  astnuiomical  tables  wiailtl  not  be  Justiliable.  The 
more  I  have  relUM-tcd  upon  the  subject  the  uuue  stronifly 
Hcems  to  me  the  evidence  that  no  such  yroup  can  exist,  and, 
indeed,  that  whatexer  anomalies  exist  cannot  be  due  to  the 
action  of  unknown  masses  of  nuitter. 

Besides,  the  six  elements  of  sim'Ii  a  j;roup  wouhl  constitute 
a  complication  in  the  tabular  theory. 

On  the  other  hand,  it  did  not  seem  to  me  best  that  we  sluudd 
wlutlly  reject  the  possibility  of  some  abn(u;nnil  action  ov  some 
defect  between  the  assunu'd  relations  of  the  various  quanti- 
ties. What  I  tlnally  dt'cided  on  doin;;  was  to  increase  the  theo- 
retical motion  of  each  periheli(Ui  by  the  same  fraction  of  the 
mean  motion,  a  cours(>  which  will  represent  the  observations 
without  committing  us  to  any  hypothesis  as  to  the  cause 
of  the  excess  of  motion,  tlnuigh  it  accords  with  the  result  of 
Hall's  hypothesis  of  the  law  of  y:ravitation;  to  reject  entirely 
the  ii)  p(»thesis  of  the  action  of  uidvuown  nnisses,  and  to  adopt 
for  the  elements  what  we  mi^iht  call  coujpromise  values  between 
those  reached  by  the  preceding  adjustment  and  thos«'  which 
would  exist  if  there  is  abnormal  action.  The  exijjency  of  hav- 
iuiX  to  prejtarc  the  tal)les  reipiircd  me  to  reach  a  conclusion  on 
this  subject  before  the  tinal  revision  of  the  preceding  discus- 


nr>,m\ 


Fl  tiki:   DKTKKMINATIONM. 


nr> 


HJoii,  so  that  the  iiiimlH'is  iisnl  an*  not  wliolly  based  upon  it. 
The  coiirliisioiis  I  have  rcachtMl  an*  these: 

SjiM-e,  if  there  is  notliiii);  altiioriiial  in  tlie  tlieoiy,  the  sohit 
paialhix  is  probably  not  intieli  hir^er  tlian  H".7HU,  anil  It  there 
is  anything  abnonnal  it  is  probalily  as  hirt;e  uh  S".7!)."»  or  even 
8".HtMt,  we  may  adopt  the  vahie  H".7!»0  as  one  whii-h  is  almost 
certainly  too  lar);e  ou  the  one  hypothesis  and  too  snnill  on  the 
other,  and  \vhi«h  is  tlu'ret'ore  best  adapted  to  alVord  a  decision 
ol  tlie  qnestion. 

For  the  nmss  ol  Venus  I  t«M»k,  as  an  intermediate  value, 

IM'  =  l-;-|O.S(MM> 


For  the  uiaas  «)f  Men-ury  I  t«M»k 

1  -  C.tNNMHNi 

Actually  it  seems  that  this  masH  is  larp'r  than  the  most  prob- 
able one  on  either  hypothesis,  tliou^di  not  without  the  ran^i'  ot 
easy  possibility. 

With  tlu'se  values  the  outstanding;  dill'erence  between  theory 
and  observatiiui  in  the  centennial  motion  ot'  the  node  of 
Venus  18 

Jsin/I>,  "  =  0".L'5 

If  this  dilVerence  arises  wlndly  from  the  error  of  the  theory, 
then  between  the  transits  of  ISTI  and  L'(M)4  the  accumulated 
error  woidd  amount  to  0"..'i2  in  the  heliocentric  latitude,  and 
about  i)".S  in  the  p>o<'entric  latitude.  Unless  an  improvetnent 
is  made  in  the  nu>thod  of  determining^  the  position  of  N'entis 
by  observation,  the  twentieth  century  must  approach  its  end 
before  this  dilVeri'iice  can  l»e  dete«ted. 

liearinfi  of  futKrt'  (ictiriniimtionn  on  tin-  qu(»ti<m. 

80.  The  Ibllowin^;  shows  the  inlluence  whicii  sul>sn|uent 
determinations  <»f  the  principal  elements  will  have  upon  our 
Jud^nuMit  a.s  to  the  solution  of  the  dilemma.  The  changes  in 
the  second  column  will,  by  emphasi/.iiij;  the  discordance 
between  tin  restdts,  tend  to  conlirm  the  hypothesis  of  an 
abnorr  ..'  .!'>feet  in  the  theory,  while  the  opposite  ones,  in  the 
latit  column,  will  tend  to  reconcile  theory  and  observation: 


IMAGE  EVALUATION 
TEST  TARGET  (MT-S) 


k 


A 


A 


I/. 


fA 


1.25 


us 


1^     IM 

2.0 


M  11 16 


V] 


y 


^> 


* 


4"^ 
^ 


Photographic 

Sciences 

Corporatiori 


23  WeST  MAIN  STMIT 

WEBSTER,  N.Y.  14580 

(716)  872-4503 


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IM 


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176 


h'  L  T I  UK   1JETERM1^' AT10N8. 


186 


Element  or  (luantity. 

Change  tending  to 
contiim  the  dis- 
cordance 
between  theory 
and  oiiservation. 

Change  tending  to  I 
reconcile  exist- 
ing theory  with 
observation. 

The  solar  parallax. 

Increase. 

Diniinution. 

Longitude  of  the  node  of  Mercury. 

Increase. 

Diniinution. 

Longitude  of  the  node  of  \'enus. 

Increase. 

Diminution. 

ConsliUU  ii(  aberration. 

Diminution. 

Increase. 

Mass  (if  \enu,- 

Increase. 

Diminution. 

Mass  iif  Mercury. 

Diminution. 

Increase. 

Secular  diniinution  of  the  obliquity. 

L -     -                               

Diniinution. 

Increa.':e. 

Amoii};'  these  constants  are  some  the  values  of  which  can 
scarcely  lie  decisiveJy  obtained  except  by  observations  con- 
tinued through  half  a  century,  or  even  through  the  wliole 
twentieth  century,  unless  improvements  are  nnule  in  our  pres- 
ent methods  of  observing. 

The  improvement  of  others,  however,  is  (piite  within  the 
reach  ci'  the  astronomy  of  the  present  time.  Among  these 
the  constant  of  iiberration  and  the  solar  parallax  havi'  the 
first  place.  The  more  accurate  determination  of  these  quanti- 
ties thus  assumes  an  importance  which  may  justify  some  sug- 
gestions on  the  subject.  ' 
.  The  observati(ms  made  on  the  European  continent  for  the 
detection  and  study  of  the  variations  of  latitmle  have  been 
executed  with  such  precision  that  Ave  might  look  to  them  for  a 
marked  improvement  in  the  determination  of  the  constant  of 
aberration,  were  it  not  for  a  single  circumstance.  In  the  gen- 
eral average  few  are  nmde  after  midnight,  while  the  maxima 
and  minima  of  aberiation  occur  in  the  morning  and  evening. 
The  extension  of  the  system  into  the  early  morning  therefore 
seems  desirable.  Although  these  observations  may  scarcely 
equal  in  accuracy  those  made  by  Nyri^.n,  with  the  prime 


801 


FlITl'KE   DETEinilNATIONS. 


177 


vertical  transit,  they  liave  tlie  iulvaiit.iyc  of  n(»t  reriniring  >^» 
louj;-  a  period  for  a  coiiiplotc  observation.  'Die  yivai  <lisa<l- 
vantaj;o  of  tlu'  prime  vertical  instrunicnt  is  tliat  unless  a  star 
culminates  within  a  few  minutes  of  the  zenitli,  an  liour.  or 
even  several  hours,  will  be  required  for  the  c<mii>leti<»n  of  a 
determination,  which  may  thus  be  made  impossible  by  the 
advent  of  daylight,  it  may  be  remarked  in  this  connection 
that  the  northern  latitudes  of  tlie  ICnropean  observatories  are 
Itivorable  to  the  determinatioji  of  tlie  alx-rration-constant. 

Loewy's  method  has  over  all  others  the  great  advantaf,'e  of 
being  iiulependent  of  the  direction  of  the  vertical.  15ut  its 
ap]>lication,  and  the  reduction  of  the  observations  made  with 
it,  are  laborious  in  a  high  degree. 

So  far  as  practical  astronomy  has  yet  develoi)ed,  the  best 
nicl  .»d  of  directly  measuring  planetary  jtarallax,  and  there- 
fore the  only  one  to  be  considered,  is  that  of  (Jill.  It  there- 
fore seems  desirable  that  measures  by  this  method  shouhl  Ije 
continued.  At  the  same  time  it  is  very  necessary  that  the 
spectra  of  the  small  planets  to  be  used  should  be  carefully 
studied  i)hotometrically,  and  that  the  probable  inlluence  of 
coloration  upon  ihe  measures  should  be  investigated. 

The  necessity  of  completing  the  present  work,  and  of  pro- 
ceeding immediately  to  the  construction  of  tables  founded 
upon  the  adopted  elements,  prevent  the  author's  awaiting  the 
mature  judgment  of  astronomers  up<m  the  end)arrassing  «jues- 
tious  thus  raised.  The  regret  with  whi(;h  he  accepts  this 
necessity  is  weakened  by  the  consideration  that  even  if  the 
solar  parallax  which  he  has  adopted  reipiires  the  laigest  i-or- 
rection  to  which  it  can  reasonably  be  sup])Osed  subject,  namely, 
one  of  — 0".(H.^,  reducing  the  value  of  this  constant  to  8  '.TT."*, 
the  effect  of  the  error  will  not  be  prejudicial  to  the  astronomy 
of  the  immediate  future. 

INIore  important  wdl  be  the  error  0".03r)  in  the  constant  of 
aberration.  Yet  a  lougcontinued  series  of  observations  will 
be  necessary  to  establish  even  the  existence  of  such  an  error, 
and  should  it  prove  detrimental  in  any  astronomical  work  the 
evil  will  be  easily  remedied  by  a  slight  correction. 
5690  N  ALM 12 


;| 


;r 


CHAPTER  IX. 


DERIVATION  OF  RESULTS. 


Ulterior  corrections  to  the  motionn  of  the  perihelion  and  mean 

longittide  of  Mercury. 

87.  In  §§.'32  and  40  we  have  reached  three  values  of  the 
correction  to  the  tabular  motion  of  the  perihelion  of  Mercury. 
Of  these  the  first  rests  on  meridian  observations  alone,  the 
second  on  the  combination  of  meridian  observations  with  trans- 
its, and  the  third  is  derived  by  substituting  in  the  eliminating 
equations  the  corrections  to  the  solar  elements  and  their  secular 
variations  which  result  from  observations.  The  three  values 
thus  reached  are  —  0".54,  — 1".01,  and  +  0".34.  The  pro- 
gressive divergence  of  these  values,  taken  in  connection  with 
the  discrepancy  pointed  out  in  §33,  leads  us  to  distrust  the 
influence  of  the  meridian  observations  upon  the  motion  of  the 
perihelion.  Under  these  circumstances  I  deem  it  advisable  to 
make  such  flual  corrections  to  the  motions  in  n-ean  longitude 
and  mean  anomaly  as  will  best  satisfy  all  the  observed  transits 
over  the  disk  of  the  Sun.  In  doing  this  I  am  enabled  to  intro- 
duce the  results  of  a  preliminary  discussion  of  the  transits  of 
1891  and  1894.  By  combining  the  observations  of  these  two 
transits  with  those  of  the  older  ones  I  derive  the  following 
values  of  the  functions  V  and  W  defined  in  §  31 : 


// 


V  = -1.93- 3.03  T 
W= +  1.50  + 2.04  T 

The  preliminary  theory,  so  far  as  yet  investigated,  gives  for 
the  values  of  this  quantity, 


V  = -2.44- 3.40  T 

W  =  4- 1.38  + 1.30  T 


178 


87,  88] 


rEUllIELlON  OF  MERCURY. 


179 


Equatiug  these  values  to  tbe  corresponding  linear  functions 
of  the  corrections  to  I,  n,  and  their  secular  motions,  we  have 
the  equations, 

//  // 

0.72  61  +  0.28  cJtt  =  +  0.12  +  0.08  T 
+  1.40      -0.49       = +  0.51  + 0.37  T 

We  find,  from  these  equations. 


ff 


// 


61  = +  0.20  + 0.50  T 
67t=  -  0.24  +  0.97  T 

The  preliminary  values  to  which  these  corrections  are  appli- 
cable are 

//  // 

61  = +0.04- 1.33  T 
(5;r  =  +  5.83  + 0.34  T 

The  definitive  values  thus  become 


// 


// 


6.    = +  0.30- 0.77  T 
(J;r  =  +  5.59  +  7.31  T 

Definitive  elements  of  the  f out  inner  planets  for  the  epoch  1850,  as 
inferred  from  all  the  data  of  observation. 

88.  We  have  made  a  fourth  solution  of  the  normal  equations 
which  give  the  corrections  to  the  elements  of  each  planet  by 
substituting  in  those  equations  the  definitive  values  of  all  the 
other  quantities,  including  the  values  of  the  secular  variations 
derived  from  theory.  In  making  this  substitution  for  Mercury, 
however,  the  ulterior  '  orrections  just  found  were  not  applied. 
The  values  of  the  unknowns  resulting  from  tlis  solution  are 
shown  in  the  first  column  of  the  next  table.  From  these 
numoers  are  derived  the  definitive  elements  for  '850,  by  the 
following  processes: 

(a.)  By  multiplying  the  unknowns  by  the  appropriate  factor 
given  in  §  27,  we  have  the  corrections  of  the  tabular  elements 
at  the  mid-epoch  of  observations  for  each  planet.  These  cor- 
rections are  found  in  the  second  column. 

{fi.)  The  preceding  corrections  are  to  be  reduced  from  the 
respective  mid-epochs  to  1850.    This  reduction  is  found  by 


180 


DEl'lMTIVE   (QUANTITIES. 


[88 


inultiplyiny  the  definitive  correction  t<)  the  tabular  secular 
variation  by  tiie  elapsed  interval,  and  is  shown  in  the  tliird 
eohunn. 

(;')  We  next  have  tlie  vahie  of  the  tabular  elements  for  the 
lundaniental  epoch  18.50,  Jiinuary  0,  (Ireenwich  mean  noon. 
These  numbers  are  those  of  Leveuriek's  tiibles,  with  the 
following  modihcations: 

(rt)  The  reduction  from  LSoO,  January  1,  i'aris  noon-,  to 
January  0,  (Ireenwich  noon 

(0  The  corrections  to  Lev^euuieu's  values  of  Ihc  eccen- 
tricity and  perihelion  which  are  necessary  to  represent  those 
terms  in  the  i)erturbationsof  the  mean  lon<iitude  which  depend 
cnly  upon  the  sine  aud  cosine  of  the  mean  anomaly.  The 
theory  is  more  symmetrical  in  form  when  all  such  terms  are 
included  with  those  of  the  ellipti(!  motion.  In  liEVERKii:K's 
tables  they  have  the  followinj;  values: 


Mercury;  (iv 
Venus; 
Earth; 
Mars; 


//  // 

+  0.030  sin/ -0.111  cos  i 
+  0.010  4-  0.037 

_  0.007  -  0.098 

+  1.001  +  0.718 


These  terms  of  the  h)njiitude  may  be  represented  by  the  follow- 
ing- corrections  to  the  elements: 


Mercury;  6e  =  -\-  0.058 
Venus;  -0.012 

Earth ;  +  0.0.")1 


Mars: 


+  0.013 


// 

dn 

.-= 

0.0 

+ 

2.3 

+ 

1.4 



1.0 

Applying  these  corrections  rf  and  f  to  Leverrier's  tabular 
quantities,  we  have  the  values  of  the  tabular  elements  as  given 
in  the  tburth  column.  Then  applying  the  preceding  correc- 
tions we  have  the  detinitive  values  given  in  the  last  column. 

In  some  cases  this  derivation  is  moditied.  Instead  of  using 
the  correction  to  the  perilielion,  mean  longitude  and  mean 
motior.  of  Mercury  given  by  the  unknown  quantities  of  the 


88] 


ELEMENTS  FOK    1850. 


181 


equations,  we  have  used  the  vahies  for  1850  derived  from  t)ie 
discussion  of  tlie  lu'eeedinfi*  section. 

The  ([uantities  wliich  j^ive  the  position  of  the  node  and 
inclination  have  been  treated  in  th«'  same  way  as  th«  v  secular 
variations.  The  symbols  .1  and  X  indi<'ate  vahu  of  the 
unknown  ([uantities  rehited  to  the  corrections  of  tlie  ..ements 
J  and  N.  These  unknowns  are  tlien  clian<;ed  to  corrections  of 
the  elements  by  the  factors  of  §1*7,  and  these  ayain  to  ('(trrec- 
tion  of  the  inclination  and  node  by  the  equations  of  §41. 

In  the  case  of  the  node  of  Venus  iwo  values  are  {;iven.  The 
value  {a)  is  that  which  follows  imnu'diately  from  the  uornnd 
equations.  If  we  carry  forward  the  position  of  the  node  Just 
derived  to  the  mean  epoch  of  the  last  two  transits  of  Venus, 
we  liiul  a  <liscrepancy  amotintinj;'  to  2".{\4:  in  the  lonj;itiide, 
correspoudinj>' to  a  difference  of  0".1lM  in  the  heliocentric  lati- 
tude. This  is  considerably  laiger  than  the  ]>ro])able  error  of 
the  results  of  the  observations  of  the  transits.  It  may,  there- 
fore, be  <|uestioned  whether  the  latter  are  not  entitled  to  a 
greater  relative  weijjht  than  that  assigned,  owin;^'  to  the  prob- 
able systematic  eirors  of  the  meridian  observations.  A  second 
value  [h)  has  therefore  been  derived  from  the  observations  of 
the  transits  alone.  In  subsecpient  investigations  we  may 
choose  between  these  two  values. 

Formation  of  drtinitive  elements  of  the  four  inner  pkmets,  for  tlii\ 
epoch  11^50,  January  0,  Greenwich  mean  noon. 

Mercury. 


Unknown  of     Corr.  of 
equations.      element. 


Rcl.  to 
1850, 

// 

o.o 


Tabular 
element. 


Concluded 
element. 


//  // 

538  100  (554.40   538  100  053.72 


n    -.0940     -    0.77 

e    -  .0741     -    0.222  -  0.005  42  400.088  42  408.861 

75  7    13.78     75  7    VX37 

323  11  23.53  323  11  23.83 

7  0   7.71   7  0  7.00 

40  33   8.03  40  33  12.24 


jt  +  .0703  -I-  5.59     0 

I  —.0402  +  0.30     0 

i  -  .2702  J  -  0.04  ~  0.07 

d  —  .0001N+  3.88  -  0.27 


f 

»    ' 

^ 

J 

1  '' 

i 

182  DEFINITIVE   QUANTITIES.  [88,89 

Formation  of  dejinitive  elements,  etc. — Continued. 

Ven  lift. 


Unknowuot 

C'OIT.  of 

Ked.  to 

T.altiilar 

Conchuled 

equations. 

clt'lMCllt. 

1850. 

element. 

element. 

// 

// 

// 

// 

n    -  .1783 

-    3.57 

0      210  669  165.04 

210  669161.47 

e    +  .1403 

+    0.43C 

)  -  0.105 

1  411.522 

1411.796 

71   +  .0835 

+  36.6 

-16.4      129 

27 

14.3 

0 

129 

27    34.5 

I    -.1330 

-    0.67 

+  0.46     243 

57 

44.34 

243 

57    44.13 

i    +  .0008  J  +    0.31 

+  0.12        3 

23 

34.83 

3 

23   35.26 

e{a)+  .0120  ]S 

-    9.39 

+  6.63      75 

19 

52.21 

7o 

19    49.45 

^(6) 

-  20.36 

+15.56      • 
Earth. 

// 

75 

19   47.41 

n 

-1.10 

129  602  767.84 

129  002  766.74 

e 

+  0.12 

3  459.334 

3  459.454 

7t 

-2.4 

100 

21 

43.4 

100 

21    41.0 

e 

-  0.15 

23 

27 

31.83 

23 

27   31.68 

I 

+  0.02 
// 

99 
Mars. 

48 

18.72 
ft 

99 

48   18.74 

// 

n   -  .1094 

-0.88 

0        68  910 105.38 

68  910 104.50 

e    -  .1088 

-  0.155  +  0.058 

19  237.101 

19  237.004 

TV  +  .1063 

+  2.38 

+  0.02     333 

17 

52.47 

333 

17   54^87 

I    -  .4029 

-0.81 

+  0.05       83 

9 

16.92 

83 

9    16.16 

i    -  .0507  J  +  0.18 

-  0.01         1 

51 

2.28 

1 

51     2.45 

0   +. 1 135  N+ 0.56 

+  1.34       48 

23 

53.02 

48 

24     0.92 

Definitive  values  of  the  secular  variations. 

89.  The  definitive  values  of  tiie  secular  variations,  as  inferred 
from  the  adopted  theories  and  the  concluded  values  of  the 
masses,  are  shown  in  the  following  table,  which  gives  in  detail 
the  parts  of  which  each  quantity  is  made  up. 

The  first  four  lines  of  the  table  give  the  values  of  the  secular 
variations  as  they  result  from  the  investigations  foun4  in  Vol. 
V,  Part  IV,  of  the  Astronomical  Papers,  after  correcting  the 
mass  of  each  planet  by  its  appropriate  factor. 

The  motion  of  the  perihelion  first  given,  denoted  by  Dt  Ttt, 
is  measured  along  the  plane  of  the  orbit  itself.    The  numbers 


891 


SECULAR    VARIATIONS. 


183 


given  being  divided  by  the  corresponding  valnes  of  the  eccen- 
tricity we  have  the  motion  of  the  perilielion  itself  along  the 
plane.  The  symbols  /„  Ji"d  ^o  represent  the  inclinations  and 
longitudes  of  the  nodes  referred  at  each  epoch  to  the  ecliptic 
and  equinox  of  1850,  regarded  as  fixed.  The  motions  of  these 
elements  are  next  to  be  referred  to  the  fixed  ecliptic  of  the 
date.  So  referred,  they  are  designated  as  D','  i  and  D;'  fi.  The 
transformations  to  the  latter  (pumtities  are  nnide  by  comput- 
ing an  approximate  value  of  the  motion  of  the  node  due  to 
the  motion  of  the  ecliptic  alone  along  the  plane  of  the  orbit 
regarded  as  rtxed. 
If  we  put 

i„  the  inclination  of  the  fixed  orbit  of  the  planet  at  any  epoch 
To  to  the  moving  ecliptic  at  any  time; 

^1,  the  longitude  of  the  corresponding  node,  Q  i ; 

V,  the  distance  from  the  node  Q  i  to  the  instantaneous  rota- 
tion axis  of  the  orbit  at  the  epoch  To ; 


we  shall  have 


Dt  V  =  h"  cosec  i'l  sin  (L"  —  di) 


{a) 


If  we  compute  vo  and  h  from  the  ecpiations 

H  sin  vo  =  sin  lo  D?  fti 

H  cos  Vo  =  D?  <o 

and  then  find  Jv  by  integrating  the  value  (a)  of  Dtv  from  1850 
to  the  date  we  shall  have 


8iniD;;^  = 


«sin  {vo+  '^v) 
«cos  (ko  -f  ^v) 


The  change  of  Dt^  between  1850  and  the  extreme  epochs  has 
been  found  so  nearly  uniform  that  it  was  sufficient  to  multiply 
its  value  a.  ohe  mid-epoch  (1075  or  1975)  by  2.5  to  obtain  Jr. 

Next,  we  have  the  changes  in  i  and  6  due  to  the  motion  of  the 
ecliptic,  represented  by  DJ  i  and  DJ  6*,  and  computed  by  the 
formula 

D{  t  =  -«"cos(v"-^) 
sin  t  D{  6^  =  —  «"  cos  i  sin  (v"  — •  ^) 


184 


DEFINITIVE   C^LANTITIES. 


[89 


I 


Tlic  pliiiu'tary  inercssion  (\\\v  to  tlio  motion  of  the  ecliptic  is 
here  omitted,  to  l)e  alterwards  iiichuhMi  in  the  general  preces- 
sion. Tiu^  snni  of  tin'  two  motions  f'ives  the  actual  variation 
at  each  ejtoch,  referred  to  a  tixed  e(|uinox. 

The  motion  of  ^'  itself  (hiis  found  is  increased  by  the  g«'neral 
precession,  which  yives  the  motion  of  H  at  each  epoch. 

The  motion  of  the  perihelion  to  be  actually  used  in  the  tables 
is  eipial  totlie  motion  of  the  node  from  the  mean  e(|uinox.  plus 
the  increase  of  the  arc  of  the  orbit  between  the  node  and 
l)erihelion.  Tiie  adopted  value  of  this  «|uantity  is  found  by 
incrcasinii'  tiic  motion  of  tti  by  the  following  (luantities: 

1.  The  clianjic  dn»'  to  the  motion  of  the  plane  of  the  orbit. 

2.  The  ehanye  due  to  the  motion  of  tiie  ecliptic. 
The  formuhe  for  these  two  quantities  are 

(l);ry,  I),  ;r=       tan  i  ;,Rin  /  D';  ^ 
(2) ;  rfi  1),  TT  =  h"  tan  i  /  sin  {L"  —  6) 

.'?.  The  excess  of  motion  shown  by  observations  in  the  case 
of  .McHMuy  and  Mars,  and  computed  for  all  four  planets  aw  if 
they  gravitated  toward  the  JSun  with  a  force  proportional  to 
y-n  ^vhere 


w  =  2.000  0001 0120 


The  >'alu 


this  correction  are 


// 

Mercury ; 

Dt 

71 

=  43.37 

Venus; 

10.98 

Earth; 

10.45 

Mars; 

5.55 

4.  The  general  precession. 

5.  In  the  case  of  the  Earth,  the  motion  arising  from  the 
action  of  the  Moon,  of  which  the  amount  is 

Dt  7t"  =  7".08 

But  the  first  two  corrections  drop  out  in  this  case. 

The  preceding  transformations  of  the  secular  variations  are 
made  with  the  original  values  of  the  elements  e  and  i,  as  given 
iu  Astronomical  Papers^  Vol.  V,  Part  IV,  pp.  337, 338. 


891 


SECULAR  VAlllATIONH. 


is: 


Secular  rariationn  <>/  the  ckmcniH  of  the  Join-  orbits  at  flic  tlmr 
epochs,  IHOO,  1850,  anil  3100,  as  inferred  from  the  (lefinitivclij 
adopted  masses. 

Mercury. 


1600. 


18.iU. 


I'll  10. 


// 

// 

// 

Jhe 

4- 

4.257 

+ 

4.227 

+ 

l.lt»0 

t'l),  Ti 

+ 

lOO.r.L'4 

+ 

I(M.».41>8 

+ 

10!).  175 

i);'/n 

— 

21.581 

— 

21.508 

— 

21.551 

9iir/„U','^o 

— 

54,8! » I 

— 

54.'.>(J0 

— 

55.04!  > 

D;i 

— 

21.78(i 

— 

21.508 

— 

21.347 

sin  i  l);  H 

— 

54.813 

— 

54.1  MJ9 

— 

55.i;{0 

d;/ 

4- 

28.884 

+ 

28.;{33 

+ 

27.785 

sill  i  D;  6 

— 

37.l!)(i 

— 

37.3J>7 

— 

37.5!>5 

I),/ 

+ 

7.()!)8 

+ 

0.705 

+ 

0.438 

sin  /  l>t  ^ 

— 

\yi.m\) 

— 

1)2.3(J0 

— 

!  12. 7  25 

J  I),  T 

— 

l.OG 

— 

1.00 

— 

1.00 

IX  ^ 

i 

>5{»3.4l 

551  ►8.70 

. 

')004.O2 

D,e 

42()2.!>.S 

42()(;.12 

42(m.24 

Venus. 


// 

// 

1  r 

Dte 

— 

!).95!) 

— 

!>.8(50 

— 

!>.772 

cDtTT, 

+ 

0.384 

+ 

0.219 

+ 

0.000 

D't'to 

— 

2.484 

— 

3.071 

— 

3.(55(5 

sin  «o  D't'  6q 

— 

59.005 

— 

59.112 

— 

5!  ►.229 

Dli 

— 

3.04!) 

— 

3.071 

— 

3.091 

sin  i  D';  e 

— 

58.978 

— 

5!>.112 

— 

59.2(50 

D]i 

+ 

0.0!>0 

+ 

0.(5!>5 

4- 

(5.097 

sin  i  D{  6 

— 

40.758 

— 

4(5.582 

— 

4(5.413 

Dt  i 

+ 

3.(541 

+ 

3.(524 

-1- 

3.(5(M5 

sin  i  Dt  f^ 

— 

105.730 

— 

105.094 

— 

105.(573 

^BtTT 

— 

0.3(5 

— 

0.37 

— 

0.38 

Dt;r 

)()90.07 

5072.44 

5054.!>2 

Dt^ 

< 

3230.39 

t 

3237.!)8 

. 

3245.22 

l.S<»  DKI'INITIVE   llEHl'I/i'S.  |81>,  DO 

8<Tul((f  raruttionit  of  the  chmeufH  of  the  four  orbitn,  etc, — ('(Jiit'd. 

IJarth. 


l(i(X). 

1850. 

2100. 

// 

// 

// 

D,  (■>' 

-       8.4«;7 

— 

8..59.-) 

— 

8.727 

e"]\  n" 

4-     1J>.2!»3 

+ 

19.210 

4- 

19.1.39 

I),  n"  . 

0179..58 

W87.41 

01!»5.08 

h"  sin  L„ 

4-       4.370 

4- 

.5..'U1 

+ 

0.305 

n"  cos  L„ 

-     47.113 

— 

40.838 

— 

4(».550 

log  h" 

l.(}7.500 

1.07340 

1.07187 

L'u 

174O42'.04 

1730  2!>'.08 

172°  17'.18 

L" 

171°  12'.83 

1730  2!>'.08 

175o40'.02 

i>o 

.')034.01 

5030.13 

5037.30 

p 

5018.28 

3023.82 

5029.38 

U,f 

-     40.701 

— 

40.838 

— 

40.847 

Mars, 

// 

// 

// 

D,  t' 

+     18,77.5 

+ 

18.706 

+ 

18.623 

cDt  ;ri 

4-  148.033 

+ 

148.707 

+ 

148.702 

Dl'  /o 

-     28.994 

— 

2!>.;{90 

— 

29.803 

sin  /oDl'^o 

-     34.023 

— 

34.012 

— 

34.017 

I);  / 

-     29.482 

— 

29.390 

— 

29.309 

sin  i  D;'  ^ 

-     33.60.J 

— 

34.012 

— 

34.445 

DM 

+     20.904 

+ 

27.104 

+ 

27.245 

sin  i  D;  ^ 

-     38.800 

— 

38.551 

— 

38.247 

Dti 

-       2..J18 

— 

2.292 

— 

2.004 

sin  i  Dt  (9 

-     72.40.5 

— 

72..5()3 

— 

72.092 

JDtTT 

+       0.08 

+ 

0.07 

+ 

0.00 

Dt;r 

0021.51 

0023.90 

062«».25 

Dt^ 

2770.39 

2770.87 

2770.03 

ISeciilar  acceleration  of  the  mean  motions. 

90.  The  mean  motious  of  the  planets,  like  that  of  the  Moon,  are 
subject  to  a  secular  acceleration  arising  from  the  secular  vari- 
ations of  the  elements  of  the  orbits.  The  following  formulie 
for  this  acceleration  are  formed  by  dift'ereutiatiug  the  known 


if^,^. 


DO 
il. 


90] 


SECULAR   ACCELERATIONS. 


187 


expressioiiH  for  the  variation  of  the  loiij-itiKh-  of  the  v\»h\i  in 
the  theory  of  thr  variation  of  elements,    the  notation  is  that 
of  AstroHomiettl  rapcrs,  Vol.  V,  Cart  I\'. 
Wo  C'lnpute  for  the  aetion  of  an  onter  on  an  inner  phmet: 

B  =i(I)-I)'-2I)')(''," 

4 

W=  i(2-UD  +  3I)2  +  41)^)rI' 

o 

Then 

D?  k  =  »i'  (*  n  I>,  I A  0-2  -f  lif'^  -  Ce"  +  \V(r'  com  (n-  -  t')  I 
For  the  action  of  au  inner  on  an  oi't^"  phmet  we  conipnte 

A'=-(l  +  D)rT 

B'  =    I  (I) +  2 1)2  4-  i)J)(''; 

4: 

0'   =      J(3D4-r.iy^  +  21)^)rr 
o 

W'=     ^(10  +  3D-1)I)^-  tl)^)c'l' 
8 

D?  /„  =  m  n'  Dt  \  A'  o'  +  B'e'  +  C'e'^  +  W'^c'  cos  (;r  -  n')  \ 

The  symbol  Dt  indicates  the  secuhir  variation  of  the  expres- 
sion following  it  prodnced  by  the  action  of  all  the  planets.  The 
unit  of  time  must  be  the  same  one  in  which  n  is  expressed. 

The  following  table  gives  the  results  of  this  conipntation: 

Secular  change  of  the  centennial  mean  motions. 

Action  of—  Mercury. 
// 

Venus,  -0.0426 

Earth,  -0.0029 

Mars,  +0.0003 

Jupiter,  -0.00.39 

Saturn,  -0.0004 

Total,       -0.0495        +0.0090        -0.0403        +0.0169 


Venna. 

Earth. 

Mars. 

// 

// 

// 

.     .     . 

-0.0104 

+  0.0010 

+  0.0128 

•         t        t 

+  0.0119 

-0.0001 

-  0.0012 

•         •         • 

-0.0046 

-0.0308 

+  0.0004 

+  0.0015 

+  0.0021 

+  0.0036 

II 

ill 
Hi 


ii 


188  DEFINITIVE   QUANTITIES.  [91,92 

The  measure  of  iime. 

91.  The  fictitious  mean  Sun  whose  transit  over  any  meridian 
detines  tlie  moment  of  mean  noon  on  that  meridian  is  a  point 
on  the  cehistial  ^pliere  having  a  uniform  sidereal  motion  in  the 
phuie  of  the  Eai  th's  equator,  and  a  Right  Ascension  as  nearly 
as  may  be  e(iual  to  the  Sun's  mean  longitude.  If  we  put  /<  for 
tills  uniform  sidereal  motion  and  add  to  ja  the  precession  of  the 
e(|uinox  in  llight  Ascension  we  have  i'ov  the  mean  Kight  Ascen- 
sion cf  this  fictitious  mean  Sun 

T  =  To  +  /<  T  +  4G0()".;}()  T  +  1".394  T« 

From  §§  88,  90,  and  100  the  expression  for  the  Sun's  mean 
longitude,  att'ected  by  aberration,  is  found  to  be 

L  =  279047'  o8".2  +  129(;0270()".74  T  +  l".089  T^ 

Equalizing  the  «'oet1icients  of  T  we  find,  for  the  mean  Right 
Ascension  of  the  lictitious  mean  Sun 

r  =  2790  47'oS".2  +  129(»0270()".74  T  -|-  1".394T^ 

This  differs  from  the  mean  longitude  of  the  actual  Sun  by  the 
quantity 

r  -  L  =  0  '.30r>  T^  =  0«.020  T^ 


It 


This  difference  is  of  no  importance  in  the  astronomy  of  our 
time,  but  may  result  in  an  error  of  2*  in  the  course  of  one  thou- 
sand years  in  the  measurement  of  time  by  the  actual  mean 
sun.  We  must  leave  to  the  astronomers  of  the  future  the 
(juestion  how  best  to  meet  the  (juestion  thus  arising.  Chang 
ing  to  time  the  expression  for  r,  the  ditterence  or  mean  excess 
of  sidereal  over  mean  time  for  the  meridian  of  Greenwich 
becomes 

T  =  18i>  :i\V"  11«.880  +  24"  0'"  1«.84449  t  +  0«.0929  T« 

t  being  time  in  Julian  years  after  1850,  January  0,  Greenwich 
mean  noon. 

Constant  ofaherratiov. 

92.  We  first  investigate  certain  fundamental  constants  con- 
nected with  the  motion  of  the  Sun,  Earth,  and  Moon,  on  which 
the  precession  and  nutation  depend. 


I; 


92,93]  MASS  OP  THE  MOON.  189 

From  the  adopted  value  of  the  tiolar  parallax, 

n  =  S".790, 
and  the  adopted  velocity  of  light  in  kilometers  per  second, 

V  =  290  800, 
follows  for  the  constant  of  aberration  the  value 

A  =  2U"..">01 

But  if  we  accept  the  mean  result  of  the  solutions  of  §  83  as 
giving  the  most  likely  value  of  the  solar  parallax,  we  shall 
have 

n  =  8". 7854 

Then  §  75  will  give 

A  =  2(V'..'511 

as  the  adjusted  value  of  the  constant  of  aberration. 

Mass  of  ike  Moon. 

r3.  By  means  of  the  e«|uation  of  §  71  between  the  lunar 
inequality  P  in  the  motion  of  the  Earth  and  the  mass  of  the 
Moon 

/<'r  =  [1.78207]  ;r 

we  may  find  a  fresh  value  of  the  Moon's  nuiss  from  the  values 
of  TV  and  P. 

We  have  found  from  observation 

P  =  0".4()5  i  .015 

Thus  follows,  for  the  mass  of  the  Moon,  when  n  =  8".7no, 

//=  1  :81.32  4  0.20 

Combining  this  with  the  value  found  from  the  constant  of 
nutation, 

/<  =  1  :  81..58  rj   0.20 
we  have,  as  the  definitive  mass  of  the  Moon, 

/I  r=  1  :  81.45  ±  0.15 


!! 


190 


W 


m 


DEFINITIVE  QUANTITIES. 
Parallactic  inequality  of  the  Moon. 


[94, 95 


94.  From  the  transformation  of  Hansen's  lunar  theory  in 
Astronomical  Papers,  Vol.  I,  it  may  be  concluded  that  the  solar 
parallax  and  the  parallactic  inequality  are  connected  by  the 
relation 

IM.  =  [1.10242]  L:^;r 

=  [1.15176]  TT 

Hence  we  have,  for  the  coefticient  of  the  parallactic  inequality 
of  the  Moon,  corresponding  to  tt  =  8".790, 

124"  .60 

Here  the  inequality  is  that  in  ecliptic  longitude. 

The  centimeter-second  system  of  units. 

95.  There  are  certain  methods  in  physics  by  which  the  next 
step  in  the  course  of  our  researches  will  be  guided.  Tlie  adop- 
tion of  a  system  of  absolute  units  has  simplified  the  methods 
and  conceptions  of  physics  to  such  an  extent  that  we  may 
find  it  advantageous  to  introduce  a  similar  system  into  those 
investigations  of  astronomy  which  are  closely  connected  with 
that  science. 

The  fundamental  units  most  widely  adopted  are  the  centi- 
meter as  the  unit  of  length,  the  gram  as  the  unit  of  mass, 
and  the  second  as  the  unit  of  time.  There,  is,  however,  an 
insuperable  ditticulty  in  the  way  of  introducing  the  gram, 
or  any  other  arbitrary  terrestrial  unit  of  mass,  into  astronomy, 
from  the  fact  that  the  astronomical  masses  with  which  we  are 
concerned  can  not  be  determined  with  sutticient  iirecision  in 
units  of  terrestrial  mass.  It  is,  therefore,  quite  common  in 
celestial  mechanics  to  regard  the  unit  of  mass  as  arbitrary, 
and  to  multiply  this  arbitrary  unit  by  a  factor  which  will 
represent  its  attractive  force  upon  a  unit  particle  at  unit  dis 
tance.  The  introduction  of  this  factor  is,  however,  needless. 
It  is  simpler  to  adopt  the  course  of  Delaunay  and  many  other 
writers,  and  regard  the  unit  of  mass  as  t  derived  one,  based 
on  the  units  of  time  and  length,  by  defining  it  as  that  mass 
which  will  attract  an  equal  mass  at  unit  distance  with  force 


. 


i 


95, 96 1 


MASSES  OF  THE   EARTH   AND  MOON. 


191 


unity.  In  this  definition  the  unit  of  force  retains  its  iihysieal 
meaning,  as  that  force  which,  acting  on  unit  mass,  will  pro- 
duce a  unit  of  acceleration  in  a  unit  of  time. 

The  number  of  fundamental  units  is  then  reduce<l  to  two, 
those  of  time  and  length,  and  the  unit  mass  becomes  a  derived 
one  of  dimensions. 

The  centimeter  as  a  unit  of  length  would  be  inconveniently 
small  for  astronomical  purposes,  if  we  had  to  deal  mainly  with 
natural  numbers,  but  it  causes  no  inconvenience  in  logarith- 
mic computations,  and  has  the  advantage  of  being  assimilated 
directly  to  the  centimeter-gram-second  system  in  physics. 
We  shall  therefore  adopt  it,  expressing  our  results,  however, 
in  terms  of  other  units  whenever  convenience  will  thereby  be 
gained. 

I  shall  make  clear  this  assimilation  and  the  use  of  the  unit 
of  mass  as  a  derived  one,  by  calling  this  the  centimeter- 
second  system. 

In  the  latter  the  definitions  of  units  in  the  centimeter 
gram-second  system  will  remain  unchanged,  except  that,  the 
derived  unit  of  mass  must  be  substituted  for  the  gram.  The 
dimensions  of  units  in  the  centimeter-second  system  will  be 
found  by  making  the  above  substitution  for  >M  in  the  expres- 
sions for  those  of  the  centimeter-gram-second  system. 

Masses  of  the  Earth  and  Moon  in  centhiuter  second  unitn. 

90.  A  fundamental  (juantity  in  the  centimeter  second  system 
is  the  mass  of  the  P]arth.  This  mass  will  be  by  definition  the 
force  of  gravity  of  the  Earth,  if  concentrated  in  a  i)oint  at  the 
distance  of  one  centimeter.  Were  the  Earth  a  sphere  of  know  n 
dimensions,  it  could  be  readily  determined  through  the  force 
of  gravity  at  any  point  on  its  surface.  This  being  not  the  case. 
we  shall  proceed  on  the  accepted  approximate  theory  that  the 
geoid  is  an  ellipsoid  of  revolution,  and  that  the  force  of  gravity 
at  a  point  the  sine  of  whose  latitude  is  1  :  -y/S,  is  th«'  same  as 
if  the  mass  of  the  Earth  were  concentrated  in  its  center. 

The  determination  of  this  constant  with  astronomical  preci- 
sion is  a  difficult  and  we  might  say  hitherto  an  insoluble  prob- 


102 


DEFINITIVE  Q  UANTITIES. 


[96 


leni,  owing  to  the  heterogeneity  of  the  Earth  and  the  absence 
otMeteriiiinations  of  the  force  of  gravity  over  the  surface  of  the 
ocean.  Although  the  limits  of  uncertainty  thus  arising  can 
not  be  set  with  any  approach  to  precision,  1  do  not  think  they 
are  such  as  to  greatly  ini])air  the  astronomical  results  which 
are  to  be  derived  from  them.  Investigations  in  geodesy  not 
being  practicable  in  the  present  work,  ]  have,  nminly  from  a 
study  of  the  work  of  G.  W.  Hill,*  assumed  for  the  lejigth  of 
the  seconds  pendulum  at  the  point  the  sine  of  whose  latitude 
is  1  :  y/'i^  which  1  shall  call  the  mean  latitude, 

L,  =  99.2715 

With  this  we  may  compare  IIelmekt's  expression  for  the 
length  of  the  seconds  pendulum  in  terms  of  the  latitude 


which  gives 


L  =  0"'. 990918  (1  +  .005310  sin  ^<p) 


L,  =  09.2C88 


From  these  values  of  L,  we  have: 

Hill.  Hklmert. 

Gravity  at  mean  latitude,               979.770  97".). 745 

Correction  for  centrifugal  force,          2.200  2.200 

Attraction  of  the  Earth,                  982.030  982.005 


I  alvso  accept  as  the  result  of  Clarke's  investigation  of  1880, 

E(iuatorial  radius  of  the  Earth,      6378249'" 
Keduction  to  mean  latitude,  7245 

Mean  radius  of  the  Earth,  G371004 

From  Hill's  and  Helmert's  numbers  follows: 


Logarithm  uuiss  of  Earth  expressed  in  centimeter-second  nnits. 


Hill. 
20.600541. 


Hel:meht. 
20.600530. 


Jatronomical  Papers,  Vol.  Ill,  p.  339. 


96,97 


PARALLAX  OF   THE  MOON. 


193 


From  the  adopted  ratio  of  the  mass  of  the  Moon  to  that  of  the 
Earth : 

fx  =  l'.  81.45 

follows 

Logarithm  of  the  mam  of  the  Moon  in  i^entimeter-seeond  units, 

18.08965. 

Parallax  of  the  Moon. 

97.  From  these  results  the  distance  of  the  Moon  and  the 
relation  between  the  mass  and  distance  of  the  sun  follow  in  a 
very  simple  way.  By  the  formuhe  of  elliptic  motion  it  follows; 
that  when  we  put 

m,  m',  the  masses  of  any  two  bodies  revolving  around  each 
other  in  virtue  of  their  mutual  gravitation ; 

a,  the  semimajor  axis  of  the  jelative  orbit,  which  would 
be  the  actual  distance  if  the  motion  were  circular; 

n,  their  mean  angular  motion  in  unit  of  time; 

we  have  the  relation 

a^  n'-  =  m  +  m' 

This  relation  is  rigorous  and  independent  of  the  adopted  units 
of  length  and  time,  provided  we  deftne  the  unit  of  mass  in  the 
way  already  done.  It  follows  that  if  the  Moon  in  its  revolu- 
tion around  the  Earth  were  not  subject  to  disturbance,  its  mean 
motion  in  one  second,  and  its  distance  expressed  in  centimeters, 
would  be  connected  by  the  relation 

Log  a^  n^  =  log  w"  ( 1  +  //)  =  20.605841 

In  the  theories  of  Delaunay  and  Adams  the  quantity  a,  as 
determined  by  this  eiiuation,  is  accepted  as  a  fundamental 
element,  and  it  is  sliown  that  in  consequence  of  the  perturba- 
tions produced  by  the  Sun  the  constant  Tin  of  the  Moon's  hor- 
izontal  parallax  is  connected  with  a  by  the  relation 

rt  sin  77o  =  1.000907/9 

p  being  the  radius  of  the  Earth  corresponding  to  Uo 
5690  N  ALM 13 


I 
w 


r 


194 


DEFINITIVE  QUANTITIES. 


[97,  98 

From  tbe  mean  sidereal  motion  of  the  Moon  in  a  Julian 
century 

1330 .  85136  revolutions 


I 

!  ' 


til 


we  find,  for  the  co-logarithm  of  the  motion  in  arc  in  one 
second 

lo*r  A  =5.574841 


and  thus  have  for  the  undisturbed  mean  distance  of  the  Moon 
in  centimeters 


and  hence 


log  a  =  10.585174 


log  sin  77o  =  8  219921 

/    // 

77o  =    57  2.68 

Red.  to  sine,  —  .16 

Constant  of  sin  rr  in  arc,        57  2.52 

Using    Helmert's   length  of  the  seconds  pendulum  we 
should  have  found  for  this  constant 


3422".55 

Masft  and  parallax  of  the  Sun. 

98.  In  the  case  of  the  motion  of  the  center  of  gravity  of  the 
Earth  and  Moon  around  the  Sun  the  relation  of  §97  beco'ies 

a''^  n'^  =  M,  +  m"  (1  +  ;<) 

Ml  being  the  mass  of  the  Sun.  Repl.acing  a'  by  tt,  the  parallax 
of  the  Sun,  and  p  the  radius  of  the  Earth,  we  And  for  the 
ratio  M  of  the  mass  of  the  Sun  to  the  sum  of  the  masses  of  the 
Earth  and  Moon 


M 

log  M  ;r' 


/)3n'2 


m"  (1  +  jw)  sin''  tt 
8.349674 


-1 


98,  99J 


sun's  mass  and  parallax. 


195 


The  values  of  M  corresponding  to  certain  values  of  the  meau 
equatorial  horizontal  parallax  of  the  Sun  are  as  follows: 


n 


M 


8.780 

330514 

8.785 

329951 

8.790 

329388 

8.795 

328827 

8.800 

328206 

Kuiation  and  mechanical  ellipUcity  of  the  Earth. 

99.  Regarding  the  mass  of  the  Moou  as  known,  we  now 
utilize  the  equations  of  §  67  to  obtain  the  constant  of  nutation 
and  the  mechanical  ellipticity  of  the  Earth.  The  last  two  of 
these  equations  give,  for  the  absolute  precessioual  constant, 
when  the  Julian  year  is  the  unit  of  time, 


P  =  [[5.975052]  j^^  +  5310".o] 


C-A 

C 


We  have  found,  in  §  66,  for  a  Julian  year 

p  =  54".8990 
We  then  have,  for  the  mechanical  ellipticity  of  the  Earth, 


C-A 
C 


0.0032753 


We  also  have,  from  the  first  equation  of  §  66,  for  the  constant 
of  nutation  for  1850 

N  =  9".214 

For  the  parts  of  the  precessioual  constant  which  arise  from 
the  action  of  the  Sun  and  of  the  Moon,  respectively,  we  have- 


Action  of  the  Sun     . 
Action  of  the  Moon . 


•         •         • 


17.3919 
37.5071 


g. 


196 


DEFINITIVE   CiUANTlTIES. 

PrecenHion. 


[100, 101 


100.  In  order  to  develop  the  terms  of  tlie  precession  and 
obliquity  to  higher  powers  of  the  time,  I  h.ave  extended  their 
computation  one  step  backward  and  forward  from  the  three 
fundamental  epochs,  by  extrapolation  of  h  and  L.  The  results 
are  as  follows : 

Motion  of  ihe  ecliptic  and  equator. 


Year. 

loj,'.   K 

L 

Dt« 

n 

o               / 

// 

// 

1350 

1.67060 

168  56.13 

-  46.013 

2009.05 

1600 

1.67500 

171  12.84 

-  46.761 

2006.92 

1850 

1.67340 

173  29.68 

-  46.838 

2004.79 

2100 

1.67187 

175  46.63 

-  4(J.847 

2002.66 

2350 

1.67039 

178    3.50 

-  46.789 

2000.52 

Centennial  precessions  for  tropical  centuries. 


m 


I 


111    lUII^llllliU  — 

In  Right 

fear. 

Lunisolar. 

Planetary. 

General. 

Ascension. 

// 

// 

// 

// 

1350 

5033.58 

-  20.94 

5012.64 

4592.41 

1600 

5034.80 

-  16.63 

5018.17 

4599.38 

1850 

5036.02 

-  12.31 

5023.71 

4606.36 

2100 

5037.25 

-    7.98 

5029.27 

4613.35 

2350 

5038.49 

-    3.67 

5034.82 

4620.32 

From  these  v.alues  we  have  the  following  general  expres- 
sions : 


Annual  precession  in  Right  Ascension; 
Annual  precession  in  longitude; 
Centennial  priHjession  in  longitude; 
,    Total  precession  from  1850; 


46.0636  I  0.0279  T 
50.2371  +  0.0222  T 
5023.71      +  2.218   T 
5023.71  T  4-  1.109   T« 


Mean  obliquity  of  the  ediptic. 

101.  The  expression  for  the  mean  obliquity  when  T  is  counted 
from  1900  is- 

e  =  23°  27'  8" .26  -  46".845T  -  0".0059  T*  +  0".00181  T' 


101,  101' 1  PRECESSION. 

Tublen  of  the  mean  oblufuittf  at  different  epochs. 


Year. 


Obliquity. 


Year. 


Obli(|uit.v. 


197 


KiOO 

2;i 

20  28.00 

-  2500 

23  58  44.00 

lono 

20  5.;u 

-  2000 

55  .38.90 

1700 

28  41.91 

-  1500 

52  23.10 

I7r)0 

28  18.51 

-  1000 

48  57.70 

1800 

27  55.10 

-  500 

45  24.14 

1850 

27  31.68 

0 

41  43.78 

1000 

27  8.20 

+  500 

37  57.97 

1050 

20  44.84 

1000 

34  8.07 

2000 

20  21.41 

1500 

30  15.43 

2050 

25  57.08 

2000 

20  21.41 

2100 

23  25  .'U.5G 

2500 

23  22  27.37 

Behitive  positions  of  the  equator  and  ecliptic  at  differtnt  dates. 

102.  The  motions  expressed  in  the  precodiiig  tables  are,  for 
the  most  part,  purely  instantaneous  ones,  referred  to  the  planes 
of  the  ecliptic  and  equator  of  each  separate  epoch.  For  the 
reduction  of  the  places  of  the  fixed  stars  from  one  epoch  to 
another,  it  is  necessary  to  know  the  relative  position  of  the 
planes  of  the  e(iuator  or  ecliptic  at  the  two  epochs.  We  shall 
therefore  derive  the  fundamental  quantities  which  express 
the  position  of  the  equator  and  the  ecliptic  at  any  one  epoch 
relatively  to  their  positions  at  a  fundamental  epoch  taken  at 
pleasure.  The  latter  we  shall  call  zero  position.  Then,  the 
zero  equator  and  ecliptic  are  those  of  the  fundamental  epoch ; 
the  equator  and  ecliptic  siniply  those  of  any  other  varying 
epoch.  So  far  as  convenient,  and  as  conducive  to  ease  in 
comparing  our  results  with  former  ones,  we  shall  use  the  nota- 
tion of  Bessel. 

To  derive  the  equations  for  the  motions,  let  us  consider  the 
following  four  points  of  the  celestial  sphere: 

Eo,  the  pole  of  the  zero  ecliptic. 

E,  the  pole  of  the  actual  ecliptic. 

Po,  the  pole  of  the  zero  e'.uator. 

P,  the  pole  of  the  actual  eciuator. 


198 


DEFINITIVE  QUANTITIES. 


1102 


Ij 


If 


We  put, 

(\  =PEo,  the  obliquity  of  th6  equator  to  the  zero  ecliptic; 

k  =EEo,  tlie  indinatiou  of  the  two  ecliptics; 

77n,  the  longitude  of  the  node  of  the  ecliptic  on  the  zero 
ecliptic,  measured  from  the  zero  equinox  of  the  date; 

/7|,  the  longitude  of  the  same  node,  measured  from  the  actual 
e(|uinox; 

A,  the  arc  of  the  equator  intercepted  between  the  two  eclip- 
tics, or  the  planetary  precession  on  the  equator; 

»/•,  the  total  lunisolar  precession  on  the  zero  ecliptic  from 
the  zero  ei)och  to  the  actual  epoch; 

w,  the  rate  of  motion  of  the  pole  of  the  oqu.itor; 

T,  the  time,  expressed  in  units  of  250  years  from  the  zero 
epoch  to  any  other  epoch. 

The  position  of  the  variable  point  E  is  detlned  by  the  quan- 
tities I-  and  Tin  or  77|,  which  are  themselves  to  be  determined 
through  the  values  of  h  and  L  of  §  100. 

The  position  of  the  variable  point  I*  is  determined  by  the 
condition  that  its  motion  is  constantly  at  right  angles  to  the 
arc  EP,  and  its  velocity  measured  on  the  arc  of  a  great  circle 
is  given  by  the  e(j[uation 

ds 


at 


=  n  =  P  sm  e  cos  e 


(a) 


The  positions  of  the  equator  and  equinox  relative  to  the 
zero  equator  and  ecliptic  are  then  determined  by  the  quanti- 
ties fi,  ip  and  \.  The  spherical  triangle  P  Eq  E  gives  the  follow- 
ing equations: 

sin  X  _  sin  77i  _  sin  TIq 

sin  k 


sm  f| 


sin  e 


During  a  period  of  several  centuries  the  quantities  k  and  A  are 
so  small  that  no  distinction  is  necessary  between  them  and 
their  sines.    We  may  therefore  put 


A  =  fc  sin  77j  cosec  f  i  =  k  sin  77o  cosec  e 


(ft) 


We  also  have,  from  the  law  of  motion  of  the  pole  of  the 

equator, 

Dt  fi  =  «  sin  A 

Dt  ^'  =  n  cos  A  cosec  €\ 


lOL'l 


MOTION   or   THE    K(,>U.\T()R. 


109 


As  tlio  value  of  f,  does  not  eliaii|,'o  by  {)"A\  from  one  epoch  to 
anotlier,  we  may,  without  apiJieciable  error,  use  f„  for  f,  in  tlie 
formuhe  (b)  and  (c).  To  u.se  tliese  equations,  we  tlrst  obtain  A- 
and  /7|  from  the  secular  motion  of  the  ecliptic,  while  n  is  com 
puted  for  any  epoch  from  the  formula  (a).  We  then  easily 
develop  the  values  of  f,  and  i/:  in  powers  of  the  time  by  the 
ec|uations  (c).  The  values  of  n  have  no  reference  to  any 
special  coordinates.  From  the  table  ot  §  KM)  it  will  be  seen  that 
we  may  put 

n  =  2004".70  -  2".13  t' 

t'  being  counted  from  IS.")!). 

To  And  the  value  of  77,  in  each  case,  we  remark  that  the 
instantaneous  values  of  L  given  in  §  100  show  that  the  instan- 
taneous node,  or  intersections  of  two  consecutive  ecliptics,  • 
moves  with  so  near  an  approach  to  uniformity  that  wc  may 
take  for  the  actual  node  between  the  ecliptics  of  any  two 
epochs  Ti  and  T2  the  mean  of  the  instantaneous  nodes  for  those 
two  epochs.  For  example,  let  it  be  required  to  find  the  value 
of  77,  for  the  node  of  the  ecliptic  of  2100  on  that  of  1850.  We 
have 


For  2100 

For  ISaO,  referred  to  eq.  of  2100 
Concluded  value  of  /7i       ... 


0         / 
L  =  17.")  46.()3 
L   =  17(;  5!).13 
77,  =  170  22.9 


As  the  basis  of  our  work  we  have  computed  the  required 
quantities  for  the  zero  ecliptics  of  1000,  1850,  and  2100, 
respectively.  The  values  of  k  and  77,  for  the  ecliptics  of  two 
hundred  and  fifty  years  before  and  after  these  epochs  are  as 
follows : 


Zero  epoch. 

-2SoY 

+  250  V 

i- 

n, 

/t 

n, 

1600 
1850 

2IC0 

// 

-118.48 
-  118.07 
-117.64 

0       / 

i6»  20.0 
170  36.  7 
172  53-4 

// 

4- 1 18. 07 
+  117.64 
+  "7-23 

0      / 

174    5-9 
176  22.9 
178  39.9 

1 

200 


DEl'INITIVK  tiUANTITIES. 


[102 


Chaii)>;iiig  tlio  unit  of  time  to  two  liiiiidrcMl  and  titty  years, 
th«  eipiatioHH  (a)  {h)  and  (c)  give  the  Ibllowing  values  of  the 
derivatives  of  ^i  and  if". 


D,e 


D,^ 


Zero-opooli. 

—  250  V 

+  260Y 

-  250  V 

+  250Y 

// 

// 

// 

// 

1(100 

-  1.4630 

+  0.7400 

I2600.;);j 

12573.0r. 

IHfiO 

-  1.17ti8 

+  0.4r)l>7 

1200H.44 

i2r)7(i.or> 

21(K) 

-  0.8898 

+  O.KMir) 

12000.57 

12579.71 

lin 


^•) 


1.3 


I: 


At  the  respective  ei)Oolis  Drfi  vanishes,  and  Dr*/'  has  the 
value«»  of  the  lunisohir  precession  in  longitude  (§  100). 
Developing  in  powers  of  r  we  have- the  following  results: 

Zero-epoch.  c      /        //  //  // 

IGOO;   f,  =  23  29  28.09- +  0.5509  r«  -  0.1200  t'-> 
1850;   f,  =  23  27  31.08  +  0.4074      -  0.1207 
2100;   f,  =  23  25  34.50  +  0.2041      -  0.1200 

//  // 

1000;    //;  =  12587.00  t  -  0.07  t« 
1850;    ?/'  =  12590.05     -0.70 
2100;    »/•  =  12593.14     -0.72 

//  // 

1000;    A  =  45.28  t  -  14.83  r^ 
1850;    \  =  33.52     -  14.8(5 
2100;    A  =  21.75     -  14.88 

These  values  of  Si  and  tp  completely  fix  the  position  of  the 
ecjuator  at  the  time  t  relative  to  the  zero  ecliptic  and  e([uinox. 
For  the  reduction  of  coordinates  from  one  epoch  to  another 
we  must  express  the  position  of  the  equator  at  the  time  r.  We 
consider  the  triangle  P  Eo  Po,  of  which  the  sides  and  opposite 
angles  are  designated 

Sides,  fo  fi  ^ 

Opposite  angles,     90°  -  C        90°  -  Ci        '/> 

If,  in  the  Gaussian  relations  between  the  parts  of  this  triangle, 
we  put 

sin  ^  (ei  -  €o)  =  ^  (fi  -  eo)  =  J  Je 


lOL'l  MOTION   OF  THE   El^lATOU.  201 

uiul  regard  the  co-sine  of  this  aii;;le  us  unity,  we  hiive 

tau  i  (;:  4-  ;,)  =  von  A  (*i  4-  '(.)  tan  A  f 
tau  i  C  -  :,)  =  2l4iiTTTf7-f-  7jlaiTT7' 

If  we  develop  the  dift'erenoos  between  the  tangent  and  the 
arc  we  find  from  these  e(|iiationH 


; -I-  ;i  =  ^'  cos  A  (f,  +  f„)  (1  +  -1^  f  .sin'  fo) 


^  =  f.u'dhl.)^'-^^--^'f'^^ 


where  we  put  ^„  for  tiie  approximate  vahie  of  t  —  ^i 

For  the  iuiiliniition  ^  of  the  nu^an  e(iuator  of  the  epoch  t  to 
the  zero  equator,  we  have  tlie  eciuatiou 

gi„  ^  ^  sin  f„  sin  V 
cos  * 

and  then,  by  developing  in  powers  of  8  and  »/•,  we  find 

i/'  sin  fo  • 

^  =  "cosr  ^    -*'/•' cos' fo) 


=  //-  sin  fo  (1  +  ^  ;')  (1  -  i  f  cos^  fo) 


We  thus  find 


Zero-epoch.  //  //  // 

1(500;  C  +  Ci  =  11543.70  r  -  6.12  t'  +  0.r)7  t' 

1850;  ll.")49.44  -0.14      -f  0.57 

2100;  11555.12  -  0,1(5      -f  0.58 

//  // 

ItlOO;  C  -  C,  =        45,20  t  -  0.02  t' 

1850;  33.53  -  9.03 

2100;  21.7(5  -  JJ.04 


// 


// 


1600;      e       =    5017.30  t  -  2.(56  t'  -  0.64  t^ 
1850;  5011.97     -2.67      -0.64 

21C0;  5006.64     -  2.67       -  0.65 

To  show  the  significance  of  the  preceding  quantities,  con 
sider  once  more  the  spherical  quadrangle  Po  Eo  EP.    Let  these 


202 


DEFINITIVE   QUANTITIES. 


[102 


letters  represent  the  positions  of  the  poles  on  the  celesticil 
sphere  at  any  two  epochs.    In  this  quadrangle  we  shall  have 


Angle  Eo  Po  E 

=  90° 

Angle  P:  P  Pn 

=  90° 

SidePoP 

=  tt 

=  90°  -  C  +  A 

Let  S  be  the  position  of  a  star  on  the  celestial  sphere.  Its 
l)olar  distances  at  the  two  epochs  will  be  Po  S  and  P  S  and  its 
Eight  Ascensions  will  be  determined  by  the  angles  Po  'ind  P 
of  the  triangle  S  Po  P. 

Thus,  if  the  Right  Ascension  and  Declination  of  S  are  given 
for  one  epoch,  we  can  lind  it  for  the  other  epoch  by  the  solu- 
tion of  the  triangle  S  P  Po  when  we  have  given  the  values  of 
the  quantities  ^,  I'l,  and  Z,-\-\. 

To  ttnd  the  values  of  these  quantities  from  the  preceding 
formula,  let  T  be  the  zero-epoch,  expressed  in  calendar  years, 
and  let  r  be  the  interval  between  the  two  epochs,  taken  posi- 
tively when  the  zero-epoch  is  the  earlier  one,  and  negatively 
when  it  is  the  later  one.  We  interpolate  the  coefticients  of  t 
and  its  powers  from  the  preceding  formula  to  the  epoch  T. 
Then  by  substituting  the  value  of  r  in  the  formula  we  shall 
have  the  values  of  the  required  quantities,  and  hence  the  data 
for  reducing  the  i^osition  of  S  from  one  epoch  to  the  other. 

O 


( 


